I just had a plumber install new moen faucets in my bathroom sink. After he left, I discovered that there is a slow drip. These faucets are brand new! What could be wrong?

Well it can only be 1 of 3 things, 1st can you tell if it is hot or cold water dripping?? 2nd is the handle loose can you wiggle them and 3rd how old is the faucet I know you said it is ''New" but it could be "old" stock from his shelf in which case the gaskets/O-rings could be dried out and needs to be replaced...
if loose then on top where the "c" or "H" is that cap comes off using a finger nail or small knife point pop off caps under caps there is a screw remove it but first turn off water under sink you will a bigger nut use a adjustable wrench to snug it more then try again if still leaking remove that big nut the is a washer on the bottom that will be leaking there are many sizes so you will have to buy an assortment with mixed sizes fond the two you need remove screw on bottom put new one in,,, put it all back together...
ps if you have vinger you can soak old washer but this only short term fix, replace washer