Let's start with a definition of what a eunuch is:
eunuch (noun): a man who has been castrated, especially (in the past) one employed to guard the women's living areas at an oriental court.
The key part being that they're castrated, often at a very young age. The result is that they wouldn't develop normally, would lack any sexual urges, wouldn't be capable of growing facial or body hair, and would also likely have a childish appearance (see Wikipedia). Those are likely enough to make them visually distinguishable from whole men.
The other clue might be the nature of the building that Daenerys was entering. If it was the chambers for a woman, or a number of women, it's more likely that the guard outside would be a eunuch. The lack of sexual urges means there's no worry about them consorting with the women at all.