How should I have played this hand?

Ultimate Bet 0.01/0.02 Hold'em (8 handed) converter

Preflop: Hero is Button with 4[img]/images/graemlins/spade.gif[/img], 6[img]/images/graemlins/spade.gif[/img]. SB posts a blind of $0.01. CO posts a blind of $0.01.
UTG calls, <font color="#666666">1 fold</font>, MP1 calls, MP2 calls, CO (poster) checks, Hero calls, SB (poster) checks, BB checks.

Flop: (7 SB) 3[img]/images/graemlins/heart.gif[/img], 7[img]/images/graemlins/club.gif[/img], 5[img]/images/graemlins/club.gif[/img] <font color="#0000FF">(7 players)</font>
SB checks, BB checks, <font color="#CC3333">UTG bets</font>, MP1 calls, MP2 calls, CO folds, <font color="#CC3333">Hero raises</font>, SB folds, BB calls, <font color="#CC3333">UTG 3-bets</font>, MP1 folds, MP2 folds, <font color="#CC3333">Hero caps</font>, BB calls, UTG calls.

Turn: (10.50 BB) 7[img]/images/graemlins/diamond.gif[/img] <font color="#0000FF">(3 players)</font>
BB checks, <font color="#CC3333">UTG bets</font>, <font color="#CC3333">Hero raises</font>, BB folds, <font color="#CC3333">UTG 3-bets</font>, Hero calls.

River: (16.50 BB) 4[img]/images/graemlins/diamond.gif[/img] <font color="#0000FF">(2 players)</font>
<font color="#CC3333">UTG bets</font>, <font color="#CC3333">Hero raises</font>, <font color="#CC3333">UTG 3-bets</font>, Hero calls.

Final Pot: 22.50 BB

Results in white below: <font color="#FFFFFF">
Hero has 4s 6s (straight, seven high).
UTG has 7s Kh (three of a kind, sevens).
Outcome: Hero wins 22.50 BB. </font>

This is a situation where I'm not sure exactly how to play. I have the second or third best possible hand and someone keeps raising me. I'm not sure if I should cap the bets because I may be beat if they keep raising me. How should I play a situation like this?