Is it safe to drive from Prague to Salzburg during winter?

As a local, I would find the Prague-Salzburg route generally unproblematic, even during winter; many people from Prague use it to get to the Alps for skiing (in which case, the tricky part starts after Salzburg). There are no tall mountains on the way (700 m ASL at most), so you should be fine assuming your car has winter tyres (zimní pneumatiky in Czech; I would not consider tyre chains necessary and spiked tyres are not even a thing here). The road quality is generally good, although not a motorway all along, it is well maintained throughout the winter. Heavy snowfall is rare so most of the time the road will just be slightly wet, but if temperatures drop below zero overnight, there might be dangerous ice (ledovka in case you listen to local radio for traffic info).

Having said that, I would much rather take the train on this route; although one or two hours extra depending on the connection, they are generally comfortable and low-cost compared to renting a car. There are generally three options, each with a single change: via Linz (shortest), Vienna (also gives a chance to see Brno on the way) or Munich (goes through Pilsen if you're into brewery tours). You can look up the connection here: https://www.cd.cz/en/default.htm

Enter "Prague" and "Salzburg" into the Rome2Rio website and you'll see lots of alternatives for getting from one of these cities to the other.

While having a car can make country touring easier because you can stop when and where you want, having a car within a European city is a challenge, and usually best avoided. These are old places, with narrow streets, and limited (and sometimes expensive) parking.

If this were my trip, I'd take the train. I wouldn't be sorry to avoid winter driving on unfamiliar roads in places where I don't speak the language.

Many people drive this route, it is all main roads and there are several routes you can take. Many of the roads will be motorways but not all. And main roads in mountains will not be straight.

If your driver has experience with driving in the weather you happen to get, in the kind of landscape you will be in and your car or van is fitted out for the weather, you should not have problems. Chains for when you get stuck in snow at least.

The biggest if is your driver. Snow and icy roads are not guarantied but when they are there is need for a driver that has experienced these conditions before in a vehicle that is like the one you will be driving. Having a driving license from Washington state does not tell anything about the driving experience of the owner.

If your driver is experienced in winter driving in mountains, it might be worth it. If not, please consider taking the train instead, or a coach of course.

The only real reason to travel in your own vehicle is wanting to make stops on the way. If your only stops are in towns or cities with railways stations or stops in the bus service you prefer, you better of taking that kind of travel. Not because car travel is dangerous, but because not having to drive is likely to make the travel much more enjoyable for the driver.

Google maps predicts a 4 to 5 hour journey by car, but if you do hit winter weather that may well become much longer. Train travel will take longer, roughly 6 to 7 hours. With one change of trains.
On the positive side, those changes of trains are all in cities you may want to visit, so you could break your journey there, without having to find parking.

Bus travel will take longer than car travel as well, but is likely to be cheaper than renting and all other costs.

In a comment is mentioned that renting a van can work out cheaper. Do compare with the actual train or coach prices and remember that fuel is/can be expensive compared to what you are used to. Public transport often has cheaper rates for young people, car rentals are often more expensive if young and/or inexperienced. Or not available at all if you only have had a license for a short time.
And if you are traveling one way, consider the drop off fee on the van, which can be substantial as you will be in a different country.
On top of that may come the price of motorway tolls, done in a once a year sticker payment which may already have been paid for the car, but if not, you will have to pay it.

A few thoughts in addition to the excellent answers you already have.

• One thing no-one has mentioned so far: the driver should be happy with driving a manual transmission car.

• If you travel on first Christmas day (25th) and maybe still on 2nd Christmas day, chances are that roads and trains will be almost empty compared to traveling before or after Christmas.
  OTOH, trains may be running on Sunday/holiday or special Christmas schedule rather than weekday schedule (Czech Republic and Austria as well as Germany [in case you take a train via Germany] have 2 Christmas holidays.) This will be more important for local trains/tram/bus schedules than for the long distance connections.

• Even if your driver is accustomed to winter driving and manual transmission, don't underestimate how much more tiring it is to drive in a foreign country (and more so during the dark): other language, unfamiliar "habits" of signing, even "only" different colors of signs, traffic lights before vs. after the crossing, right before left (and that right may be lost if not used) instead of first-come-first-serve means that the driver will need more concentration than at home because many things that an experienced driver has automated won't work as automatically.
  For the Indian driver that would also include right-side driving.

• For the Christmas days, plan and read ahead what will be open when. I'd expect cities like Prague and Vienna to have some Christmas market still open, but in smaller places they probably close at 24th noon at the latest.

• I'd expect a rental car that is to go from Prague to Austria to be properly equipped with winter (or all season) tires, but it will likely not have chains. Which is IMHO not a problem in the sense that unless your driver is experienced with winter driving with chains, in case you get into conditions where you'd need chains, your best bet is to wait for the road to be cleared.

• That is one secret how people with only medium winter driving experience get safely along: patience. Stick to main roads, if the conditions become bad, stay in one of the towns alongside (the whole region is densely settled) and wait until the road clearing trucks did their work. Continue when the road is clear again. And maybe don't be too proud to walk a bit and instead save your driver from the need to go into small towns in bad weather: the small roads in towns are likely to be worse than the main highways, so better park the car where the road is good.

All that being said, it's not something that I'd consider per se unsafe to do.

If not, what are some good alternatives to travel to Salzburg apart from flying.

Assuming you are flexible, I would recommend going from Prague via Vienna to Salzburg on the train, with an overnight stop in Vienna. This way, you can see another major city.

Looking on https://tickets.oebb.at/ it takes 4 hours from Prague to Vienna with Railjet and you can buy tickets in advance -- called "Sparschiene" "saverails (railsave)" for 30 euros.

Looking on https://westbahn.at you can buy a ticket in advance for again around 30 euros and it takes around 2 hours 30 mins.

If you drive make sure you get "winter tyres" (Winterreifen). I found the motorways are generally fine. Sometimes when there is a lot of snow, it becomes slow single-file, which makes the journey longer.

I would recommend to get the train, however.

I drove this several times and to be honest, I don't understand your question "is it safe...?". Please remember one thing - going by car over mounains in winter is never safe, you never know what can happen, on the other side, there is a very good road in the mountains between Prague and Salzburg.
Summary: if you are an experienced driver, don't think about it, it's just another standard Europe road which are well serviced especially on Austria side. Otherwise, take a bus/train/flight.

As an alternative, google for "Prague Salzburg Limo". €379 gets six people and luggage in large car from Prague to Salzburg, and you can add stops on the way at sightseeing places.