Is this only due to steric factors or do electronic factors (such as ortho-para directivity) play a role here too? In particular, I suspect ortho-para electronic factors might be significant because some of the resonance structures have a cation directly next to the electron-donating group.
Both steric and electronic factors play a role in this reaction. Resonance structures such as those shown below favor electrophilic attack (the diazonium ion is an electrophile) at the ortho and para positions. Since there are twice as many ortho positions as para positions in N,N-dimethylaniline, on a purely statistical basis we might expect a 67:33 ortho/para product distribution if only electronic effects controlled the reaction. Since diazonium coupling usually produces a majority of the para coupled product, steric factors must be playing a role as well. The somewhat bulky dimethylamino group does make it more difficult for an electrophile to approach the ortho position and the diazonium ion is relatively large. These two steric factors combine to reduce the amount of the ortho isomer and the para isomer predominates.