Partially answered in comments, a summary:
You have a good idea, basically, but also other ideas can be tried.
You might however consider a model in which movies that belong to many
categories have the effects of each category count less strong. For
instance if the categories comedy, romance, drama have a certain
similar (positive/negative) effect size than this should probably not
be tripled for movies that happen to belong to all three
categories.This is especially important if movies have been
categorised with different 'intensity' of number of categories.
– Martijn Weterings
But how? Let the number of genres marked for a film be $k$. Then using a linear model with
\E Y=\mu =\beta_0+\beta_A x_A + \beta_B X_B + \dotsm
and $x_A, x_B, \dotsc$ being the indicators for genres $A,B,\dotsc$ and $k=x_A+x_B+\dotsm$ then we could replace the model above with
\E Y=\mu =\beta_0+\beta_A x_A/k + \beta_B X_B/k + \dotsm
$$ maybe, that is, replace the 0,1 indicators with $0,1/k$ indicators. Maybe you could also include $k$ (or some function of it) as a covariable. You could also try to estimate the effect of $k$, but that could lead to a nonlinear model. I would try out various possibilities.
Then another advice was
No problem, recommender systems can deal with this. How many genres do
you have in total?