As others have mentioned, they don't have a lot of purpose on modern digital cameras as far as IQ goes...protection tends to be the most common use. As long as you are getting one of good optical quality, they most likely won't cause any degredation of image quality...at least not noticeably...so it's just whether it makes you feel better having it on there. Get a cheap one though, and the IQ may suffer - making it not so worth it. Those who have smashed the end of a lens on something, or gotten a bad scratch on the end of the lens glass, would appreciate having a UV filter which would take most or all of the damage - being a simple disposable piece, it's much cheaper to replace than a lens...but I don't go out of my way to put one on my lenses.
I use one on my 50mm F1.7, because I had one with the right threads that fit, and that lens has no hood. I use one on my 300mm F4 prime, because it came with one - and it's a hugely expensive piece too - so might as well use it. All of my other lenses do not have one. Lens hoods can offer protection too, so if you use one, you don't really need the extra UV protection as well. And remember for night shots and such, any extra glass at all in front of the lens can cause unwanted reflections and flares - so it's usually a good idea to take off a UV before this type of shooting.
Sony DSLR-A580 / Sony 18-250mm / Minolta 50mm F1.7 / Sigma 30mm F1.4 / Tamron 10-24mm / Tamron 150-600mm / Tamron 90mm F2.8 macro / Minolta 300mm F4 APO
Sony A6000 / 18-55mm F3.5-5.6 / 55-210mm F4-6.3 / 10-18mm F4 / 35mm F1.8 / 16mm F2.8 / via manual adapter, lots of Pentax K mount, Konica K/AR mount, and Leica M mount manual lenses