Yes, you may prohibit visible tattoos entirely or you may simply prohibit those that are offensive, distracting, inappropriate, or over a certain size.
Tattoo policies usually depend on the culture of the workplace and are often found within a broader dress code policy. Some employers avoid restrictive dress codes because they may deter impressive job candidates from applying or drive away high-performing employees. Employers who want to maintain a certain company image, however, might prefer a strict dress code. Striking a middle path is also an option—something like “Tattoos must be appropriate and in keeping with a professional image.”
When creating your policy, make sure it doesn’t discriminate based on a protected class. This would include, for example, making religious accommodations. You should also be sure to communicate your reasons to employees and apply the policy consistently (while allowing for required exceptions).
This Q&A does not constitute legal advice and does not address state or local law.