No particular age has been set in stone, but there is some evidence to suggest as early as 18 months of age (Eichstedt et al., 2002; Poulin-Dubois, 1998). However, it is quite possible the answer depends on the child's cognitive, social, language development. In fact quite a bit of literature suggests that at the very least 2 years old's are able to associate the relevant gender stereotypes to themselves and to others ( Serbin et al. 2002; Poulin-Dubois et al., 2002 ).
However, what causes these 1 1/2 year old or 2 year old children to already have preconceived ideas on gender type? Most of it can be attributed to indirect cues that were provided by parents. To illustrate that point, Gelman, Taylor, & Nguyen did a study where mothers would read picture books to their child(ren) and found that these mothers were subtlety giving cues like "Most girls don't like trucks" and so forth (2004).
Now lets address this question are cues the only source of gender typing? No, no they are not.
The first factor is the family/parents, but I already addressed that to a certain degree. However, I wanted to point out that some parents do actively reinforce gender type. I am not going reference a study for this as I am sure most people can think of at least anecdotal evidence of this statement.... just know that anecdotal evidence is "true". Furthermore, I am ignoring siblings as they tend to play a similar role to peers which is addressed later on.
Another influence on gender typing is biology. According to de Waal, all male mammalian species activity tends to be aggressive and females prefer activities that require certain amount of emotional sensitivity and both prefer to associate with same-sex individuals {this same-sex association is important, but I'll address that a little later}(1993, 2001). It is the very likely reason why a boy might like an aggressive sport like football and why a girl might like a emotionally sensitive event like a tea party, that is, their genetic disposition makes certain activities more "pleasurable" than others.
The last factors is are your peers. Your peers reinforce gender typing. An example of such a reinforcing behavior is boys and girls will actively ignore/criticize any of their peers that engage in cross gender activities (Fagot, 1984). Consequently, due to this behavior children come to believe the legitimacy of gender segregation and gender-stereotyped activities (Martin et al., 1999). Keep in mind there are cultural difference as well, but this sums up the generality pretty well.
There are some other factors such as role modes (E.G. teachers, T.V. Stars) and just the general social environment (E.G. commercials). However, these factors tend to be in my opinion not that significant. I'll go into a little more depth if there is a demand.
Disclaimer/Notes
The most significant problem to this question is how do you measure the influence of gender stereotypes on a child who is most likely unable to communicate with you, let alone give any significant insights? In fact, if you clicked on the Poulin-Dubois 1998 link you'll notice that this article tests 18 month year old's if they can distinguish between a lady and a man which is not direct evidence. However, I think if the child is able to generally understand the concept of gender then they can be gender typed. While I have no direct evidence back that claim up I think it is accurate. Furthermore, I could easily argue that very young babies could be somewhat conditioned to certain stereotypes, but for reasons that would add another unnecessary paragraph I will excluded that argument. Feel free to disagree with either statement.
Lastly, this answer is based largely upon on some notes that I wrote a year ago in a child psychology class so if I wrote something in err let me know and I'll correct it or suggest why it is correct.
