Wasn't their concern that if free states continued to be added to the United States, but slave states were not, then it was only a matter of time until the free states had the legislative numbers in Congress to outlaw slavery?
Amendment required. So the 14 slaves states existing in 1860 would be enough even at the present day to prevent ratification.
I don't think any sane southerners in 1860 seriously believed slavery in the states was immediately threatened. I believe they were just sick and disgusted, not unreasonably, with sharing a nation with people who constantly told them the basic institution on which their society was based was evil.
The problem, of course, is that it was evil.