A tactic I often use is bribing the Senate with cash gifts to get my standing higher. What I found was this:

At the start of the gift-giving, it actually increases your standing with the senate AND the people.
A bit further in, it increases your standing with the people more and stops helping your senate popularity.
Later, it simply makes no difference.
And if you continue to bribe for a while after that, your standings with the Senate goes down, it hasn't happened with the people yet.