That of course depends on how you look at them, since "Caucasian" is not a well-defined term.

In genetic terms, the population of Asia minor and the region of Constantinople at the time of the Byzantine empire were their own thing; to nobody's surprise a 2011 study by Ottoni et al found a rural population resembling both Greeks and Persians, also sharing a considerable amount of alleles with the Levant. Notably, they found no particular relationship to Central Asian genotypes.

A slightly more recent study by Ottoni's group in 2016 establishes the sampled population as representative of southwestern Asia and the Roman core, with genetic distances (ie. "relatedness") gradually tapering off in all directions but remaining particularly high into the areas of today's Georgia, Armenia, Azerbaijan, Syria and Iran.

Interestingly, the latter study also compares ancient DNA to modern populations in the area, and found a very high degree of continuiety. This suggests that the cultural and initial social dominance of the asiatic Turks did not result in a significant population replacement, at least not in rural areas. Life most likely went on with new rulers, who eventually were absorbed into the general population.

In short, they were Anatolians and part of a continuum that is not reflected in a discourse artificially dividing humans into distinct goups as if that in any way reflected historical genetic reality. Thus, wether one call the Byzantine population of Anatolia "Caucasoid" or not says nothing informative about history, only about the preferences of the speaker.