Can I block entire genres?

Yes — Block Genres, in the Settings sub-tab of your Requests tab, lets you rule out whole categories of music at once. It's a Pro feature.

You get eleven genres as tap-to-toggle chips: Pop, Rock, Hip-Hop/Rap, R&B/Soul, Dance, Electronic, Country, Jazz, Classical, Latin and Metal. Block as many as you need. The filtering then works exactly like the explicit filter — matching tracks are removed from your guests' search results before they can pick them, and guests see a short notice telling them some results were hidden because the host has restricted certain genres.

These genre names deliberately mirror the labels used by the music catalogue your guests search, and each track is matched on the primary genre that catalogue assigns it. So blocking Metal removes everything catalogued as Metal, but a crossover track the catalogue happens to file under Rock will still come through. If you need to keep out particular songs rather than entire styles, deleting individual requests before you sync gives you finer control.