@fresh2deafbill: Live view isn't required, the optical viewfinder is actually better in most cases. When using live view, all DSLRs are very, very slow to autofocus. The opposite is true when using the optical viewfinder. The only time that you'll probably miss live view is when you need to focus very precisely on something, because most cameras allow you to blow up the image on the LCD in real time to see minute details.
As for other recommendations, you'll have to tell us how much you're thinking of spending.
I find the viewfinder a lot nicer than live view, the only time you use live view is when you have to make a picture in a hard position like above your head. So if you plan on photographing events with people it might be a good thing to have but not neccesary.
@fresh2deafbill: I just thought you were talking about the friction of a DSLR with that title...
No, you don't need Live View, it's just kind of a gimmick anyway. Define "budget". I'd consider the Nikon D3100 a starter budget camera or, for the same price you could get a Nikon D3000 with a kit lens.
I have never looked for Live View in µ DSLRs. It just doesn't seem necessary, but if you're shooting with a tripod very often than I could understand the appeal. I would advise against a Sony, by the way, as their lenses can be very expensive (it's a low buy-in system--they seem to make most of their money off of overpriced lenses). The Nikon D3000 ($499) is a great camera and is slightly more money, but looking for a used camera, such as the popular and economical D40x ($225-325) or the Canon 40D ($400-500), both prices for a kit. It's a good idea to consider lenses too, though nowadays the two companies have come to be closely matched. Understand that the brand you buy now could dictate which you use for more than a decade; If you have $1000 worth of Nikon lenses you'll be hesitant to purchase a Canon camera or vice versa. In this instance I'd recommend purchasing the used D40x kit and then buying something like the low-priced 50mm ƒ1.8 ($135).
You definitely can live without Live View. I use it only very occasionally, but I'd say over 99% of the shots I take are looking for the viewfinder.
What I would advise is investing in either Canon or Nikon,, especially if you intend to stick with photography as a hobby. The Sony Alpha series is pretty good, and they're improving all the time, but they just don't have anywhere near the breadth or depth of lens selection or lens quality. Even if you have to buy secondhand to come in under your budget, a used canon 20D is a solid workhorse and you should be able to get a camera body for under $300.
I'm glad it's on my camera, but apart from using it to help manually focus while on a tripod, I very rarely use it. Focusing through liveview is much slower than through the viewfinder, especially in darker settings due to the method of autofocus used in liveview (usually contrast measurement).