Hello Karen,
We've had a similar situation in the past couple of months not necessarily due to spambots but because valid users with JavaScript scripting disabled in their browsers. As a workaround, we create a "gotcha campaign" that checks for blanks in any of the required fields and send them back to the referral page or form, forcing them to enter all required fields in order to move to the next flow step -in our case, receive a confirmation email with the event's details or download url.
We also add smartlist filters to the actual registration campaign that checks for no blanks in all required fields so the status gets changed to either "Registered" or "Rejected" as appropriate. If they're "Rejected" - meaning a blank required field was found-, they will then get picked up by the gotcha campaign and sent back to limbo url until the enter all the required fields.
We still get some of these blanks from time to time, but at least it's easier to identify them since they're all contained within a campaign.
I do like
bgomes idea about adding the noscript tag also but we like to set up a campaign to can keep an eye on the offenders.
Hope this helps.