very slim
but if you are very concerned and anxious get tested for HIV qwith HIV P24 antigen antibody test which is very accurate

From the CDC website: Blood must come in contact with damaged tissue or be directly injected into the bloodstream (from a needle or syringe) for transmission to occur. So did the barber use an alum block *that was disinfected in between customers*? I am unfamiliar w/ the texture of alum so I think a bleach wipe may be a good way to disinfect it in between customers. Alum helps to close up the skin so, suspect the risk is low w/ a very small area of your open skin, but the volume of blood on the alum may not be zero if it's not disinfected properly and blood-borne (HIV, hepatitis C, and hepatitis B) viruses could potentially be transmitted if there is blood. I do not believe alum has antiviral properties. I think it's safest if someone has a nick to provide pressure w/ an new alcohol wipe (like what is used to clean your skin prior to a blood draw)-as a quick kill disinfectant to the skin and the small nick should close on its own and be provided a bandaid if needed. If you want to use your own alum stick/block or styptic pencil at home just for you that's fine.
(For these types of cases, w/ a small volume of potential blood and a small non-severe open area and not a hollow bore (ie blood inside) needle, no post-exposure prophylaxis/PEP for HIV is recommended since 2005 for unknown source exposures- and yours would be an unknown source. I hope this leaves your fear, however, if it does not, get HIV/Hep C tested today, abstain from any/all risky non-condom sexual encounters (oral, anal receptive, or insertive), and get re-tested again in 2-3 weeks. I assume you are immune to Hepatitis B as every US kid is vaccinated since the early 1990s. If unknown, check for hepatitis B surface antibody as well. If not immune to hep B, do get vaccinated; there is a vaccine that is new and requires only 2 doses separated by one month/ helical.

Thanks for the question. If the blade of the razor was not used previously, you are at no risk. The alum is not a problem.