The Enter key isn't really creating a separate new field -- it's just that the Enter key creates a paragraph break within the field, and the new paragraph is separated from the previous one by the Space After setting of the current style.
There are two issues here. One is that you don't want to allow new paragraphs inside a form field, and the other is that you probably don't want entries in a field to expand the depth of the table row and throw off the alignment of the form.
The first issue can be handled with some macros, which are given at http://support.microsoft.com/kb/211219. Although the article says it's applicable to Word 2000 and 2002, it's actually still valid all the way up to Word 2013 (currently).
The second issue is a bit touchier: If someone types in a long text, even without pressing Enter, the field will expand and push the rest of the table down. You can prevent the misalignment by going into the Table Properties dialog and setting the row height(s) to an Exact size. The user can still type more than will fit in the table cell, but the overflow will disappear below the bottom of the cell.
If the filled forms will be processed by macros, you may have to deal with overfilled fields, but that's another issue.
