You can short something with $1. It's just not enough to affect the value itself. You also don't technically need any money at all, the point of a basic short is to profit off selling borrowed stock after all.
You definitely do need money. No broker is going to let you short without collateral, and you're going to be paying interest for the duration of your short position beside any fees/commission.
Maybe if you have a super low cap, high fees, and they automatically close your position at a pretty conservative point. But that'd hardly be worth any broker's time with that risk/reward, unless they are hosing the borrower with insane fees. Though if that's the case, putting up collateral would be cheaper (even if you have to borrow it from somewhere).