Yes, if that is what your doctor is testing for. There is a specific blood test that can test for both Herpes Simplex I, and Herpes Simplex 2. (Typically referred to in medical communities as HSV-1 and HSV-2). In response to the virus the body produces antibodies, the blood test checks for the presence of these antibodies. If the antibodies are present for HSV-1, then you've been exposed to and most likely carry that virus, the same for HSV-2.
Both HSV-1 and HSV-2 can cause genital sores, however HSV-2 is the virus genreally associated with HSV-2. HSV-1 causes cold sores around the mouth, but as a I stated, can cause sores on the genitalia.
A doctor may recommend a culture of a sore when an outbreak occurs to double check the diagnoses, and help with treatment.
So yes, there is a bloodtest, and it can differentiate between the two strains.
Studying lab sciences