If influence was logistics, and you had a had a short hard limit outside the influence (logistics) border you could travel but that your colonization and starbase expansion would have to be inside the influence zone. Would drastically slow down expansion. Expansions would be much more ordered and developed along the meniscus of the influence (logistics) border.
Other players sending ships inside your border would carry diplomatic penalties and allow you to shoot those ship out of space without war autodeclared. Since influence is highly variable depending on the strength of the number of people and their morale and etc, the moral of the story would be that your logistics could function the same way. Only the highest strength fleets could be in areas where you can logistically support them.
Would make for a more interesting experience I think as the current no- border, broken influence, silly logistics system on a hex map is just boring.