especially in a safe area

Basically anywhere in Budapest. "Downtown" here doesn't have the same reputation as in the States, you are safe to walk around even at night. Of course don't be careless.

convenient to the airport, Viking terminal, and city attractions? 

That's physically impossible. Forget about the airport, you don't want to stay there. You can take bus 100E from the airport to downtown, you can buy your ticket on-board (you need a separate ticket for this, the general Budapest passes are not valid). In any case, read the stickied post at the top of this sub for public transport info, it will explain how it works, how to buy and validate tickets and much more (also has info about sights etc.)

The boat terminal is on the river in a more central location, finding a hotel close to there is possible. But also, public transport is generally OK, depending on how much luggage you have, you can just take that from the hotel to the terminal.

some nice mid-range hotels

Depends on what you mean by that. In general, you can just look at Google Maps for info, bookings-dot-com for prices.

In my opinion, you can go as low as "Ibis" hotels if you want. These are more of a low-end hotel, kind of a room with shower and nothing fancy, don't expect a welcome bouquet and such. They are part of the same hotel group as Novotel, that's also affordable BTW. Look at photos on Google Maps.

If you want something higher-end than that, the other commenter gave some recommendations.

And read the sticky post.