Senate influence is mainly provided by power base. You need to give powerful Oligarchs the best positions in goverment and negate the other factions powerful characters the chance to influence the Senate.
There are two positions in the Senate that give you more senate influence, the Agoranomus (+10% senate influence) and the Themostete (+15% senate influence). You want to designate the most power base oligarch in your nation in both positions. Normally, Heads of Family.
On the other hand, to negate other heads of families that are not oligarchs to have any senate influence, you should send them as legates or governors. Because this is dangerous, you should choose a small region or legion to put these heads of families so they do not become rebellious.
Finally, your admiral should also be an oligarch, as admirals do not have their senate influence negated and they can be very powerful depending on how many units they have in their command.
Themosthete Senate Influence bonus
Governor -100% senate influence
Legate -100% senate influence
Admiral Senatorial Influence based on navies
List of democrat influential characters that you have to negate
Characters by power base: the first three are heads of families you have to negate their senate influence as they are not oligarchs