OK, this is hard to put delicately, but worth reinforcing: if you have warts on your hands, you're not going to end up with warts in your more intimate regions through touching those regions. Shaking hands is potentially risky, but shaking the cage is not a problem.
Picture by Sean B
A post on The Conversation by University of New South Wales lecturer Michael Tam notes that while all warts are caused by human papilloma viruses (HPV), the strain that causes genital warts is different to that which causes those typically found on hands (and other body parts):
Warts are contagious: HPV can be transmitted by direct contact through minor injuries in the skin. After infection, there can be a latency of weeks to years, so warts can appear to come out of the blue. But rest assured, if you have a common wart on your fingers, you are not going to give yourself genital warts, or even plantar warts, if you touch those parts of your body. Different types of warts are typically caused by different types of HPV, so you could potentially infect the other hand.
The post also notes that in the case of common warts, most will disappear without treatment eventually. Popular treatments, such as duct tape, don't have scientific backing through wide-scale trials.
Monday's medical myth: warts aren't contagious [The Conversation]