It sounds like you're storing the genres as a comma-separated string. You can use the MySQL function FIND_IN_SET, i.e.:
SELECT *
FROM movies
WHERE FIND_IN_SET('adventure', genres) > 0
Another way to check a CSV field string is via the LIKE operator, like so:
SELECT *
FROM movies
WHERE ',' + genres + ',' LIKE '%,adventure,%'
We surround the genres field with commas so that the format is more regular.
Unfortunately, CSV fields don't index as easily, and your query may be slow. A better solution would be a many-to-many table of movies and genres.