First of all, knowing which version of NX you're using would really help when it comes to showing you what commands to use, as their locations can change from one version of NX to the next....or they could even be completely renamed. Regardless, please state your version of NX when posting a question.
Next, make SURE the STEP file was exported as a solid and not surfaces. Some CAD softwares allow for surfaces only in STEP files and that might be why you're looking at a surface model in NX. Also check your STEP import options in NX. Set it to import solids only unless you don't mind surfaces slipping through. If you set your STEP Import Options to solids only, then you don't have to worry about someone accidentally making a surface-only STEP file...it won't import the surfaces. Make sense?
Check your model tolerance in NX. If your modeling tolerance is tighter (smaller) than the software that exported the STEP model, then you're wasting your time trying to get a solid from the STEP file. The tolerances need to either match or NX's tolerance needs to be larger. That's just my personal opinion. Hard to convince me that a larger toleranced STEP file would import as a valid solid into a tighter tolerance in NX, but I've seen stranger things happen with CAD before.
You can use many techniques to repair the imported geometry. Some might include finding the areas with missing surfaces and using the adjacent surface edges to create a Through Curves (TC's) Through Curve Mesh (TCM's), N-Sided Surface, or any other surface command that allows you to control continuity (tangency, curvature, etc.). You might have to use Bridge Curves if the adjacent surfaces are a mess as well.
Should you find a small gap rather than missing surfaces, you can try to edit individual surfaces and untrim (Edit -> Surface -> Boundary) them back to their original shape prior to trimming then re-trim to desired boundaries. Sometimes this prior trimming information is not retained in the translation of the surfaces, other times it is.
If you can untrim, then re-trim surfaces, that will be your best option for preserving the original topology (shape), however, we don't live in a perfect world and you might have no choice other than to rebuild some of the surfaces as mentioned above. As caddict pointed out, being able to play around with the model would help us help you.
I hope some of this proves to be helpful.
Enkei America, Inc.