What is a good Lake District base for hiking?

Depends a lot of the interpretation of your question.

If you stick to a short walk from the train station Windermere is the natural option.

If you are willing to loosen that requirement a little and accept traveling by bus, the whole Lake District is a possibility. Bus transportation in the Lakes is very good, but a bit pricey (to my taste at least). Anywhere in the central Lakes you will find really nice. Most places are very easy to reach without driving yourself. At that point it depends on the kind of accommodation you are looking for. Are you thinking hotel, hostel, B&B, camping... ? Just last weekend I was bivvying out near Grisedale Tarn. It was really nice, but might not be what you are looking for. According to me, you can pick about any spot in the central Lakes. It will be nice and offer plenty of opportunities for all kinds of walks.

We have stayed at the Rampsbeck Country House Hotel. It's right on the Ullswater on the A592 south of Penrith Cumbria.

The surrounding hills are honeycombed with paths, byways, and old droves suitable for hiking and/or mountain biking. There's also a couple of farms offering horseback riding. There's a boat service that can take you down to the Trust or Information Centre in the southern part of the lake. Hence, a good place to set your 'base of operations'.

There's a sports store across the street and a small grocery store where you can stock up.

The hotel itself is pricey. In fact very pricey. But great accommodation and the Sunday roast is packed.

Transportation: the hotel is a 3 minute walk from the Watermillock Train Station and trains run daily from Penrith. By car, be prepared for arduous driving once you leave the A66, also mobile dips on the A592 that disable satnavs.

I have no connection to the hotel other than previous occasions as a customer. Trip Advisor reviews at http://www.tripadvisor.co.uk/Hotel_Review-g1096480-d191540-Reviews-Rampsbeck_Country_House_Hotel-Watermillock_Lake_District_Cumbria_England.html

Walking routes, hiking paths, and hiking information for Ullswater is at Walking and hiking around Ullswater

(Images from Cumbria Tourism, with permission from Shaun Walby, also with permission from helvellyn)