How bad does it look if I asked the director of graduate studies if I can change my program of study from pure to applied math after admission?
Assuming your personal reasons are sound, a polite conversation cannot hurt you. Explain your reasons, and keep an open attitude. It is very common for people to change sub-fields (and fields) after their PhD, or even postdocs, so it will not strike as an big oddity.

Now, if it is possible or not, depends mainly on where does your funding come from; and if the university regulations allow it. This will be brought up in the conversation.

If your reasons for studying applied maths are your desire to clone green dogs, well, it will look bad.