If you don't want to fix it, this is more handyman territory than plumber territory. Shouldn't be that expensive. However, your best bet is to at least attempt it. Worse case scenario, you can't fix it and have to buy a new faucet.

Usually when I'm attempting a DIY, I figure out make and model then do a search for it, like 'x faucet leaking at handle' then see what comes up on YouTube. Lots of good info on the major brands out there. Also if this is a higher end faucet, they are built to be repairable. The cheap $40 ones you get at the big box stores you can easily replace the gasket if it doesn't stop dripping when you turn it off, but something like the handle leaking might be a toss it and replace scenario.

I learned to replace my first washer on a faucet in my kitchen when my husband was too busy and my landlord's handyman kept blowing me off. The bedroom was off the kitchen so it was making me nuts. Took me about five minutes once I found a YouTube, watched the video, found the washers at the hardware store and a screwdriver to remove the screw. You can probably do this if you have any amount of patience. Oh and the will to clean the cabinet out, but you'd have to do that for a handyman anyway. Good luck!