The 8th Circuit, which has a few details on Earthworm Jim HD's multiplayer. According to Gameloft's Anne-Joelle Casenaz, the XBLA and PSN versions will feature "brand new levels with exclusive enemies and up to 4 players in co-op multiplayer!" So there you go: It's co-op.
According to a list of Achievements unearthed by Xbox 360 Achievements, it looks like the upcoming Earthworm Jim HD will feature some form of multiplayer. There are two Achievements for multiplayer, one awarded for being "'No. 1' in each Xbox Live multiplayer level" and another for completing all multiplayer levels. It's hard to imagine how multiplayer would work, though GamerBytes offers up the possibility of speed run competitions or a multiplayer variant of the snot bungee mini-game.
We'd also suggest that the space races between levels could work as well. Also, given the title of the second Achievement is "World Partnership," the multiplayer could be cooperative in nature.