Black wins.
Black has a distant passed pawn and Blacks pawns are more advanced. White is bound to defend against g-pawn, while Black can do anything with a and b pawns. And White cannot win a pawn race, because Black is more advanced.
[Event "Original Position"]
[Site ""]
[Date ""]
[Round ""]
[White ""]
[Black ""]
[Result "0-1"]
[FEN "8/8/8/5kp1/p7/P4K2/1P6/8 b - - 1 2"]
[Setup "1"]
2... g4+ 3. Kg3 { Option 1, defending against g-pawn, fails.} ( 3. Ke3 { Option
2, counter attack on the other side, which fails also, because Blacks pawns are
more advanced, Black wins the race.} 3... Ke5 4. Kd3 Kf4 5. b4 axb3 6. a4 g3
7. a5 g2 8. a6 g1=Q ) 3... Kg5 4. Kg2 Kf4 5. Kf2 g3+ 6. Kg1 Kf3 7. Kf1 g2+ 8.
Kg1 Kg3 9. b4 axb3
However, if Black wouldn't have a-pawn that advanced, White could catch Black and avoid promotion after dealing with g-pawn.
For example:
[Event "Queenside Pawns Shifted"]
[Site ""]
[Date ""]
[Round ""]
[White ""]
[Black ""]
[Result "1/2-1/2"]
[FEN "8/8/8/p4kp1/P7/1P3K2/8/8 b - - 1 3"]
[Setup "1"]
3... g4+ 4. Kg3 Kg5 ( 4... Ke5 5. Kxg4 Kd5 6. Kf4 Kc5 7. Ke5 ( 7. b4+ axb4 8.
Ke3 b3 9. Kd2 Kb4 10. Kc1 Kc4 11. Kb2 Kb4 12. a5 Kxa5 13. Kxb3 ) 7... Kb4 8.
Kd4 Kxb3 9. Kd3 Kxa4 10. Kc4 { Black king is trapped. Draw.} ) 5. b4 ( 5. Kg2
$2 $19 Kf4 6. b4 axb4 7. a5 b3 8. a6 b2 9. a7 b1=Q 10. a8=Q Qc2+ 11. Kg1 Qd1+
12. Kg2 Qe2+ 13. Kg1 Qe1+ 14. Kg2 Qg3+ 15. Kf1 Qf3+ { And black takes
opposition, which is necessary to promote g-pawn.} 16. Qxf3+ Kxf3 17. Kg1 Kg3
18. Kh1 Kf2 19. Kh2 g3+ 20. Kh1 g2+ 21. Kh2 g1=Q+ ) 5... axb4 6. a5 b3 7. a6 b2
8. a7 b1=Q 9. a8=Q { This is a bit complicated, but according to tablebase,
this is draw.}
6 men tablebase
and their result can be checked online here. The position is lost for White...