How would you determine the genotype of a person with a dominant phenotype if they don't wanna have sex?

You can always go to a genetic councilor who can create a pedigree (a family history of the disease) and can determine the chances of your children having the phenotype.  

I learned that in bio as well as using a testcross.  The testcross would be more accurate for the genotype, but the pedigree can help with the phenotype.

For humans, it's not so hard.  Most of human's characteristics are determined by more than one gene, the only exception I know is blood type.  So, because of this, the homozygous dominant is very rare, second only to the homozygous recessive, and they both have already been determined using statistics, so you wouldn't have to do a test cross.  For instance, in skin colour, the homozygous dominant is a very special type of black skin present in a tribe in Australia and the homozygous recessive are albinos.  The other way is to do a pedigree, which is a sort of statistical method, where you follow a certain phenotype during several generations, but you have to bear in mind that every member of the offspring followed the probabilities of being homozygous dominant of recessive or heterozygous.  The final and actually most expensive way is to perform a sort of paternity test, you take cells of the person, multiply the DNA by PCR and run a gel.  Since the gel messures the weight of the DNA, homozygous and heterozygous will weight differently (in kilo Daltons, a VERY small weight unit for DNA).