How much was "Sonnet 130" influenced by modernism?

I think you need to be very very careful with your terms
here. Let us remember that this excellent sonnet was written by Shakespeare during the
Renaissance, in sixteenth century Britain. Modernism, as a period of literary criticism,
actually is thought to have started in the late nineteenth century. Therefore, the quick
answer to your question is that this sonnet was not influenced in anyway at all by
modernism, because modernism as a philosophical approach had not even come to exist
during the time of Shakespeare, and would not for a few more
centuries.

However, it is possible to use modernism to
analyse the poem. One of the great aspects of literary theory is the way in which we can
go back, using the lenses of such approaches as modernism, to look at and analyse texts
such as this sonnet and we can see whether this approach casts any new light on the
poem. However, we need to be very careful to not confuse our time periods as you have
done in this question.