This can be a very sensitive time for both you and your neighbour. You want to help and your neighbour probably doesn't want to accept "charity". There are several things that come to my mind - not all of them concerning a gift basket.
Regarding a gift basket, maybe being a "secret" neighbour is the answer. Prepare a gift basket, wait until the coast is clear, then deliver the basket to the doorstep with a small type-written note (to maintain anonymity) that just says something like: "From a friend who cares." Many times, it's having to actually face people who know their circumstances that causes the embarassment on the part of the recipient.
I live in an area where farmers' markets are everywhere. Many times they sell items like potatoes, onions, carrots, and other good vegetables in very large quantities for a very low price. Once you get your bounty home, you can discover that you have accidently purchased too much and need to share in order for the produce to be used rather than let it go to waste. If they insist on paying you, emphasize the fact that they are doing you a favour by taking it off your hands. If they still insist, ask for payment in the form of them babysitting for you for an evening free of charge or something else that won't involve cash.
Do either of your neighbours have a talent that you could pay for? Is the husband a handyman? What about that small repair you have meant to have fixed for so long? Does she sew? How about those clothing repairs you haven't made?
Hopefully, some of these ideas will help you out.