I do not believe the Health Dept can assist you with this situation but you should call them to see. I would suggest writing the owner of the unit above you a letter specifically outlining the problems, i.e., apparent plumbing problems form the unit are causing leaks into your unit. Those leaks are causing damage and mold and the situation needs to be remedied immediately. Also, request the unit owner notify his or her condo insurance carrier of this potential claim. Date the letter and make sure you keep a copy for yourself. If you mail the letter Certified, also mail a copy by regular US Mail. Certified letters are often refused or unclaimed. Note the method of delivery in the body of your letter, i.e."Certified Mail and Regular US Mail."
I would also send a copy of your letter to your condo association. I think most condo association agreements have a clause that requires owners to undertake reasonable steps to make sure problems within their unit, i.e., water leaks, do not harm neighboring units.
If no response from the owner and/or owner's insurance company within a week, then you should seek the advice of an attorney in your area. If the mold exposure causes health problems for you or your family, you may have a claim against the owner. You may also have a claim against the owner for any property damage caused by the leak, and this is why you want the owner to notify his or her insurance company. (These claims may be covered under the general liability portion of this or her policy.)
I concur with Attorney Wolfe. The one tip I might add is to make sure you document all your efforts, take photographs, etc.
I am licensed in Massachusetts. Any advice I provide is for informational purposes only and in no way constitutes an attorney/client relationship. If you are in need of legal counsel, you should contact a local attorney.