Yes, ongoing effects (such as those from a Sea Tower) persist even if the hex gains a Curse.

This is supported by the following rule:

• On page 14, it states:
  "Ongoing effects (such as on a Sea Tower) are active as soon as the hex is øipped, even if there is a Curse."

Additionally, on page 49, it clarifies:

• "A hex with a Curse is its own type of explored terain, and loses al the properties of the original hex, except impasable terain, or ongoing effects (like on a Sea Tower)."

Summary

Ongoing effects on a hex remain active even if that hex gains a Curse.

Source: page 14, page 49

Source: Core Rulebook (pp. 14, 21, 23, 49, 55)