What are the advantages to having a majority party in the senate?

The majority party appoints the chairs of committees and controls the agenda for items coming the the floor for action.

Because the senate is part of the process of suggesting and ratifying bills, having a majority of Democrats in the Senate make it more likely that bills in favor of a Democratic idea will be passed. However, if a bill is passed by the majority party in Congress, but the president from a different party vetoes it, a two thirds voted in both the House and Senate are needed to make it a law. Those are very simple terms of putting it but it is basically the process.