Manchester United's worst finishing position in the Premier League has been third in the following seasons:
2001/2
2003/4
2004/5
In every other year since the start of the Premier League in 1992 they have finished either first or second.
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They were relegated once.
Above answer is technically not correct, question asks lowest position in the Premier League, which started in the 1992/93 season (yes they have been relegated way back in the old Division 1 days).
Liverpool's lowest finish in the Premier League was 8th, in the 1993/94 season. However, the rate they're going in this 2011/12 season, they're gonna beat that all time low!
Since the Premier League was founded in 1992, the lowest position Arsenal have finished is 12th in the 1994/95 season
Annswering myself - it looks like 15th in Division 2 in 1902 and in my lifetime (born 1948) it was 1st in Division 2 in 1975 according to Wikipedia