Other answers here cover the protection issue fairly well — there are other ways to protect your lens which work as well or better in most cases, and the loss of image quality is very real. (See Are there any downsides to using a good-quality UV filter?). But I wanted to run some risk analysis numbers.
Let's say you have a fancy $1700 lens that you're worried about damaging. Prices for a UV filter in the right size range from $7 to $420. You decide you want to avoid the cheap end, but the top seems a bit much. You look at the middle of the list... the median is about $80. You want a little better than average, so you go for a $100 model. Seems okay for peace of mind, right?
But, what's the chance that damage will occur? Glass is actually pretty strong, and modern materials and coatings make front elements rather resistant to damage. And, although not great for resale value, tiny scratches or similar don't have meaningful effect on image quality — I'm sure everyone's seen the LenRentals smashed front element example by now. I'd be inclined to guess that the chance of damaging the front element in a non-cosmetic way is less than 1% per year (unless you're in extra-dangerous conditions), but for the sake of argument, let's say it's 5%. Over the course of 10 years, that 5% accumulates to an overall 40% chance of an incident (which, again, I think is very high).
But, the cost for replacing the front element for that lens is about $200. That means the expected value with a 40% chance of needing a repair is... $80. So, why spend $100 on a filter?
Or, let's say it's the kit lens that came with your camera. The image quality concerns of cheap filters don't go down for cheaper lenses, so we're looking again at least a $500 filter — but, never mind repair: you can almost certainly buy a replacement for the whole kit lens for $50 on eBay.
This is an example of loss aversion — a natural human cognitive bias where we fear loss more than we appreciate equivalent gains. I might go so far as to say that camera stores which push protective filters as a must-have are preying on this weakness for profit.
But, of course, if you're in a dangerous environment, a filter might make sense. And some lenses have surprisingly high front-element replacement costs. These things might change your math. LensRentals actually has a more recent post on all of this which isn't a comprehensive guide but gives the price for many common lenses (and is worth a read on this topic in general) — Front Element Lens Protection Revisited.