Rather than spending gold (although that can make sense at times), you can pursue a quest strategy.
When you meet a CS (particularly one with a nearby barb camp), you can pledge to protect, which increases your base influence to 10 (will gradually rise to 10, which is 2/3 of the way to friendship). Then monitor announcements of quests -- barb camp elimination, find a civ, attach a resource to your trading network, build a specific wonder, etc.). You won't be in a position to satisfy all quests, but when you do satisfy a quest, you will get 30-45 influence points (depends on quest), so you can expect to enjoy 10-15 turns of friendship from one quest (shorter if the CS is hostile) and much more for multiple quests. You can cancel your pledge after 10 turns, to avoid minor antagonism with other civs that bully the CSs (the diplo hit is modest), but you can't re-pledge for another 10 turns.