Barring Antibody-dependent enhancement (ADE), how else can vaccines enhance the infection?
Vaccines are now well-tested to make sure that they don't do make infection worse, but there are many ways they COULD.

Two early formalin-inactivated vaccines (RSV and measles) both had this effect. See this Reddit post  to understand the mechanisms.

Theoretically, a poorly-designed vaccine could worsen viral infections  in three ways:


by biasing the adaptive immune response in the wrong direction that makes it harder to effectively respond to the actual virus.  This might be by priming the wrong type of response, e.g. producing a strongly biased Th2 response when a Th1 is required); creating the right response but to the wrong epitopes, in a way that subsequent exposures can't overcome - "original antigenic sin"; or by inducing T/B cell anergy or tolerance to the virus in question.
immunopathology.  For many viruses, the immune system's attempts to kill the virus, not the virus itself's direct cytolytic effects, damage the host.  E,g, Hepatitis C has 3 broad outcomes to a natural infection:  about 20% of people create an early, strong, multi-specific CD4/8 response that clears the virus.  The rest develop chronic infection.  In most people with chronic infection, the immune response damages little - viral loads can be extremely high but liver fibrosis is minimal.  But in \~10-20%, there is a strong immune response: chronic inflammation, infiltration of lymphocytes, and constant direct killing of virally-infected hepatocytes by CTLs leads to deposition of scar tissue and, eventually, cirrhosis.  So if you created a vaccine that failed to clear the virus and instead generated a pro-inflammatory, pro-fibrotic immune response, it would be a disaster.
Increased target cells.  This was a concern with one of the experimental HIV vaccines.  It increased the number of CD4+ cells permissive to HIV infection in the female genital tract (and the rectum, but I don't remember), thus giving the virus more target cells to infect (there was also an experimental HPV vaccine that was feared to do the same).  Those vaccinated thus had increased risk of infection.