The only way to get more boarding passes is to upgrade to premium (bronze, silver, gold or platinum). Each tier comes with a set amount of boarding passes :) 

Either buy more land to board horses at home/more stalls or build larger barns that houses more horses but takes up more land or simply upgrade. Once those passes are used up in basic you are stuck with it.

You could create up to two alts to have more room if you so wish and shuffle horses around that way, for example, I have two alts and the third one is mostly a storage bin until my main gets more room, but outside of upgrading or buying more land/stalls or tinkering with what you have already, there isn't much one can as a basic when running low on boarding passes.

Okay, thank you guys!