I actually think this is a complicated question, given it raises the main causes of attraction and correspondingly a good working definition of 'standards' is appropriate. If you think of standards within a 'attracted to' framework (not specifically sex or dating) then I actually think men hold higher standards, as their attraction circuitry is primarily visual based (i.e. the females "looks"). This is biological, however it can be viewed particularly from a liberal feminist standpoint as superficial and shallow. On attraction alone, if a woman is conventionally beautiful then she will have 'passed' the man's standard (this is not to say he would actually do anything beyond this, given that an interplay of personality etc would then come into play). For the woman though, her standard for attraction is much more fluid and distant from social rigidity. They are constantly judging, based on behavioural traits which align with her biological impressions of masculinity. This can be a variable and often confusing process, even for the female herself (given that the males 'attractiveness' can be lost very quickly at anytime in comparison to the male quickly making up his mind and the female from that point "having passed his test").
So to answer in short, if you are talking about standards in the purely physical realm, the answer is males. In a more complicated, prolonged judging sense, definitely females.