I just arrived and I didn't have to go through the search of an apartment so I cannot answer that question.
Regarding the transports, Aesch possesses quite a good connection to Basel.
You can find the travelling informations and maps on this website: http://www.bvb.ch/en/news/current-info
I personnally live along the tramway line 11, with 25 minutes connection to Basel SBB (train station).
Otherwise the Dornach-Arlesheim train station joins Basel SBB in 10 minutes every 30 minutes. The informations can be found there: http://www.sbb.ch/en/station-service...ity-basel.html
I hope it'll help you.
If there's something to know about Swiss transport, they are on time with accurate timetables.