I was about to write a post on this topic, then saw this article that pretty much sums it up:
And then this one: http://abcnews.go.com/ABC_Univision/Politics/heres-shutdown-affects-immigration-services/story?id=20431892
Let’s leave those articles to the pros, but in short, those with applications with USCIS should not be directly affected since USCIS is funded primarily by application fees. All USCIS offices are currently open and operating as normal. http://www.uscis.gov/
Visas processed through the Department of State (at consulates) should also continue to be processed. http://www.state.gov/m/rls/2013/214880.htm
Those in removal proceedings will have their court dates rescheduled unless they are detained. Detained court dates will continue to take place. http://www.justice.gov/eoir/ICstatus.htm
Don’t think that the federal government shut down is a good time to meander across the border without documents-Customs and Border Protection, and ICE are all still fully operational.