I don't see any upsides to having them. I do know that the state of Wyoming has them on their aquatic invasive species (AIS) list and that they have actually banned live possession of brook stickleback anywhere in the state.

https://gf.state.wy.us/web2011/Departments/Fishing/pdfs/AIS_STICKLEBACK_INFO130003770.pdf

Quote:
Brook stickleback have been shown to compete with and negatively affect other fish species and waterfowl. Studies show that waterfowl may be negatively impacted by brook stickleback due the species’ affect on zooplankton biomass and abundance. Brook stickleback are known to forage for other fish species’ eggs which may negatively impact fish populations and result in reduced fishing opportunities.