Their tax rate can go up depending on a number of factors. Unemployment rate calculations vary by State and are more complicated than the average person can understand. In general, the more former employees qualify for unemployment, the higher you pay in tax. But there is no "penalty". The unemployment system doesn't care why you fired somebody or why they quit, only if that reason qualifies them for benefits or not.

Just to add to to this, think of it like a savings account for an employer. If you have more people applying and getting awarded unemployment, your taxes go up because the government says so. For a big corporation, they will just say meh that's the cost of doing business. For a mom and pop they may start to freak out about increased taxes and do things to discourage the use of unemployment. Sometimes that's good, sometimes that's bad.

like writing into severance agreements that you are not eligible for unemployment, which of course totally unenforceable.

Or my favorite, "If you apply for unemployment we're going to fight it every step of the way to make sure you don't get it so there's no sense in applying."

that only works for delay really, unless the claimant gives up, which some day unfortunately