When I hire an assistant may I establish a dress code?

Yes. Employers are free to establish dress codes.

Here's what the U.S. Equal Employment Opportunity Commission says about dress codes:

Dress Code

In general, an employer may establish a dress code which applies to all employees or employees within certain job categories.  However, there are a few possible exceptions.

While an employer may require all workers to follow a uniform dress code even if the dress code conflicts with some workers' ethnic beliefs or practices, a dress code must not treat some employees less favorably because of their national origin.  For example, a dress code that prohibits certain kinds of ethnic dress, such as traditional African or East Indian attire, but otherwise permits casual dress would treat some employees less favorably because of their national origin.

Moreover, if the dress code conflicts with an employee's religious practices and the employee requests an accommodation, the employer must modify the dress code or permit an exception to the dress code unless doing so would result in undue hardship.

Similarly, if an employee requests an accommodation to the dress code because of his disability, the employer must modify the dress code or permit an exception to the dress code, unless doing so would result in undue hardship.