Extension of stay at USA due to COVID-19 lockdown

I would recommend you file for an extension as soon as you are sure you are not likely to be able to leave on time. All things considered, this will probably be early May.

Here is the current situation:

• India has suspended all commercial flights in and out of the country through 3 May 2020. By 4 May at the latest, there should be a new announcement, and you should know if the suspension remains in place or has been lifted. If the suspension gets extended into June, your flight will almost certainly be cancelled.
• In normal circumstances, your I-539 extension of stay must be filed and receipt acknowledged by USCIS before your I-94 authorized stay ends. Given the coronavirus, they are giving some flexibility for late submissions, but it's still a good idea to file early. It is currently taking several months to process these applications. You can remain in the US until your application is processed and you receive the decision, but if it is refused you must leave immediately.
• Repatriation flights are still flying, in both directions. You can contact the Indian embassy to find out if a repatriation flight is available and what the requirements are.

With regard to your passport:

• The Indian embassy in Washington has suspended passport services except for emergencies. Because it is more difficult to renew an Indian passport outside India after it has expired, you should contact the embassy as soon as possible regarding a possible passport renewal. Don't follow the normal passport instructions on the website unless instructed by the embassy as that process has been suspended.
• You can use the existing passport to return to India all the way up to 23:59 on its expiry date. But if you are not able to get a new passport and you are stuck in the US after your passport expiration, the embassy can still issue an emergency travel document to allow your return to India after flights eventually resume.

Michael's answer is very thorough, but I wanted to pop this up top:

If you want to go home now, contact your government about a repatriation flight. Governments are working hard to get citizens home.

Otherwise I would file that I-539 extension absolutely as soon as possible, and state the reason as fear of ability to get a flight home due to the virus, and a wish to keep your immigration record in good standing. The reason is the backlog of these requests.

I don't know how immigration law will treat this. My dealings with regular law is that courts seem to be treating the crisis as a tolling (stopping of the clock) on matters that have a time limit. I would hope that in the future, immigration officials will likewise "stop the clock" on evaluating overstays.