Though the NIV translates v23, "mercy mixed with fear", the literal Greek says, "mercy in/with fear".
Any fisherman will tell you that there is such a thing as a "healthy respect for the sea". This "respect" essentially boils down to being afraid (and rightly so) of what might happen to you or your boat at sea during bad weather. Often what we call "common sense" is basically a healthy fear of something: whether fear of being drowned by the sea, fear of being robbed when alone on the street at night, fear of kidney failure from picking and eating the wrong kind of mushroom, fear of getting burned by fire, or, as in v23b, fear of the garment polluted by sin.
The Bible is very clear that we must avoid sin at all costs. This does not mean that we avoid sinful people: rather, as v23b says, we show them mercy. Even though their garments are defiled by the flesh (i.e. they are stained by sin), it is our duty to help them. But in helping them, we must have a sensible fear of the sin by which they are defiled, and a godly hatred of their sin. Put in Pauline terms, when helping others out of sin, you must "consider yourself lest you also be tempted." (Gal 6:1).
Thus we see a threefold escalation in v22-23:
1) Have mercy on some, who are doubting
2) Save others, snatching them out of the fire
3) On some have mercy with fear, hating even the garment polluted by the flesh
Such threefold escalations are very common in Jude. All of this must be read in the context of the main theme in Jude: false teaching and the sin it causes and is caused by.
Summary: Be merciful to sinners, but you must fear and hate their sin, lest you be defiled by it.