Munich is probably best from the standpoint that there is at least one direct train to Budapest. However, most will require a connection in Vienna. Frankfurt is also a good option as it’s a major hub airport — you can fly directly to Frankfurt from major US hubs. The most direct trains involve a connection in Vienna. There are a couple of 1 connection options directly from FRA airport, so you wouldn’t need to go into Frankfurt at all.

As for train tickets, a eurail pass will cost more than train tickets (unless you’re doing more travel than City in Germany-Budapest-City in Germany) and may involve as much or more “hassle” (it’s very easy to buy a train ticket in Germany) as buying your ticket at the station. You will have to get your railpass validated at one office or window and probably have to buy a reservation for the specific train you want to take to use at another window — that’s more “hassle” than buying a train ticket. I would try to buy the train ticket, with reservation, at http://reiseauskunft… That will be cheapest and easiest. Look for special fares 60 days before departure.

If you want to see Germany, there are only two German cities with direct trains to Budapest: Berlin and Munich. From Munich, you will have to pass through Vienna anyway. From Berlin, you’ll have to pass through Prague and Bratislava anyway. The countryside is nice on both legs, maybe the Munich leg has less variable coutryside than the Berlin leg.

Both legs can be done overnight as well, either 9 hrs train ride from Munich (EN 463 “Kalman Imre”) or 13 hrs ride from Berlin (EN 477 “Metropol”).

To be more exact…
…if you don’t mind ONE single transfer, then it could be anything. I was bored so I looked around how to get to Budapest in the morning while leaving the westernmost point possible in the evening. Look what I found (runs daily):
Leave Paris-Est, 15:24, train TGV 9575
Arrive Munchen Hbf. 21:38
wait 2 hrs 2 minutes
Leave Munchen Hbf. 23:40 train EuroNight 463 “Kalman Imre”
sleep 8 1/2 hrs
Arrive Budapest Keleti 08:54 the next day

If you play with the trainsearch a little, you will find that with a single transfer in Vienna, or in Munich, you have practically covered the northern half of Italy, whole Switzerland, whole Austria, large parts of Germany, ex-CzechoSlovakia, ex-Yugoslavia, France, Netherlands, Denmark. It just depends how long you want to travel. The abovementioned example Paris-Budapest takes 17 1/2 hours, BUT you travel 1658 km (1030 mi!), so that even with a two-hour-pause of transfer counted in, your average travel speed is about 60 mph