Browser service in sql server clustering
There really isn't a short answer because there are a few hidden questions in the question.

A few thoughts to help here:

1.) The Browser service is not cluster aware, so it generally would be just running on each node. The browser is really used to handle incoming connections to a SQL instance. When you don't have a fixed port, are using named instances and in other situations the browser handles the "finding" of the instance a client desires to connect to. So if you have it running on the active node and it is being used to direct connections, I would make sure it is automatic and running on each node.

2.) That said the browser shouldn't prevent DLLs from being found or take any part in preventing or allowing a failover. So the issue you are having with failing over is most likely not related to the browser but something else. Instead you should be looking at things like - Have you failed over before? Is this a new install of a second node? Have there been any required restarts missed on that node? Did the install throw up any errors? Are there issues in the clustering logs? Can you post the exact error you are receiving? Have you searched for that exact error?