The influence buildings have their own tree. Its a mechanic separate from tourism. Consulate is the basic influence building for most races. While tourism gets a bonus from high influence, its not high enough to really take advantage of.Tourism doesn't get very big, but its free. Its not uncommon for a tourism building to cost more in maintenance then it gains from the tourism bonus.
Influence too has a disproportionate cost to benefit ratio. The basic influence building only provides a 5% influence bonus and planets have a base of 1 influence which can't be increased AFAIK. The building level bonus is 10% so getting bonuses on influence buildings is very helpful. Since it takes 4X influence to flip a planet, and influence tends to diminish over distance, it requires a ton of influence-producing buildings and starbases in order to push influence significantly.
This is the best way I've found to spread influence...
1. Pick a frontier planet you want to push influence from in order to begin flipping nearby planets. Turn that planet into a production center with a small ring of influence buildings and a shipyard.
2. Build 3 influence starbases around that planet. All the influence upgrades on starbases also increase influence growth on the planet. This is on top of the culture the starbases produce. Each starbase produces 1 influence multiplied by its % upgrades. This double influence boost is whats really needed to get serious about influence growth.
3. Have this planet produce culture. This high-production planet can build its own starbases, and when its not it can get a massive 200%+ additive influence growth multiplier.
4. Begin constructing a line of culture starbases as close to the enemy borderline as possible. Squeeze these starbases as tight a possible, and just keep building row after row as the borders expand. Don't worry about the upgrades unless you have excess resources. All of the upgrades in total equal half an influence point, while a single unupgraded starbase equals 1 influence point.
5. Focus on the third benevolent tree which provides a nice influence boost.