First make sure Fast Reply is enabled in your forums:
ACP > Forum Control > Manage Forums > Edit Forum Settings >Postable Forums Settings:
"Turn on the Quick Reply Box?" YES

Then each individual user can select whether to have the quick reply screen open with every topic here:

My Controls > Menu > Options > Board Settings> Board Display Settings:
"Open Fast Reply automatically when available?" YES

If you wish to force the Quick Reply box to always be open for everyone, you may install this code (fast reply must still be enabled for each forum):

ACP > Skinning & Styles > Board Wrappers > Footers:

Thanks Helena! I've been looking for this for sometime :)