How bad does a cut have to be to get tetanus shot?

Booster? Depending on the circumstances of the cut and when you had your last tetanus booster. They should be given every 10 years.
Dirty. The cut would need to be deep, ie penetrating. Dirty wounds necessitate tetanus booster. And of course the time of last vaccine is important. If greater than 5 years since last dose of tetanus.
Any skin opening. An untreated foot laceration that clogged thru the swamp, mud & dirt would be the prime illustration of tetanus risk. Yet, a simple shallow opening in the skin, contaminated by tetanus spores ( that are all around us but predominantly outside in dirt) could give you tetanus. You put yourself at risk every time you let your system weaken & the booster is simple & readily available.