To clarify, you are talking about the two eels from Disney's The Little Mermaid, and Ursela the seawitch in disguise.
If you add armature of some sort to the eels such that they hang straight, you can connect that to clear plastic rod (about 3/16'' diameter). You then mount the other end of the rod to a plate, which you mount to a belt worn under the costume. The costume would have a hole to allow the rod through. You do this on both sides, and get the appearance that the eels are swimming alongside you.
But that's sorta advanced stuff that's pretty easy to screw up. You'd need to heat-bend the rod to get natural shapes. You'd probably want to make the connection between the rod and the plate removable, so you can quickly remove the rods when in tight spaces. You'd want to be careful about any sharp edges, so no one gets hurt.
To save yourself the trouble, yeah, either safety pins, or sewing patches of velcro onto the costume and the eels is totally the way to go.