Black has various ways of declining the gambit. The most reliable are the ones that undermine White's e4-pawn, taking advantage of the fact that 4.c3 is not really a developing move. Tame ways of declining such as 4...d6?! or 4...g6?! allow White to set up a classical two-pawn centre with 5.cxd4.
This is the most popular way of declining the gambit, and the only way that definitely equalises. Correspondingly, in the Chesslive.de database, Black scores a respectable 48% with this move.
It is often played on general principles (don't go pawn grabbing in the opening, get your pieces out) but is also sometimes used with the specific aim of reaching sterile endgames to bore White into submission. The line in question was introduced by Capablanca vs. Marshall, Lake Hopatcong 1926: 4...d5 5.exd5 Qxd5 6.cxd4 Bg4 7.Be2 Bb4+ 8.Nc3 Bxf3 9.Bxf3 Qc4. Black can also reach it via 6...Bb4+ 7.Nc3 Bg4 8.Be2 Bxf3 9.Bxf3 Qc4. Black scores 58% with this line, even though it is objectively equal. Marshall tried the very risky 9.Be3, but it seems that the only reliable continuations for White are to exchange queens with 10.Qb3 Qxb3 11.axb3, or 10.Bxc6+ bxc6 11.Qxe2+ Qxe2+ 12.Kxe2.
Those who are interested in playing the Göring Gambit must have something ready against 4...d5. Maybe some may see enough play in the 10.Qb3 line above, but White also has ways to avoid these endings. After 6...Bg4 White can play 7.Nc3, and then 7...Bxf3!? 8.Nxd5 Bxd1 9.Nxc7+ leads to another ending, but a somewhat sharp and unclear one with chances for both sides. In the critical position after 6...Bg4 7.Nc3 Bb4 (or 6...Bb4+ 7.Nc3 Bg4), White has Mark Nieuweboer's suggestion 8.Be3, which appears fine, and the less reliable 8.a3 that was often used by Martin Voigt and analysed in some detail in Danish Dynamite.
Having avoided the aforementioned endings, play typically progresses into isolated queen's pawn positions which are objectively equal, but White tends to get a plus score in practice, because the white pieces often spring to life during the early middlegame. For example, after 4...d5 5.exd5 Qxd5 6.cxd4 and the sensible alternative 6...Nf6, White is scoring 56%. Black quite often goes for aggressive approaches with queenside castling, but then White can often develop a strong attack down the half-open c-file.
An interesting alternative for White is 5.Bd3!?, which is effectively a double pawn sacrifice in view of the line 5...dxc3!? 6.exd5 (6.Nxc3 dxe4 7.Nxe4 Bb4+ is better for Black) 6...cxb2 7.Bxb2 Qxd5. This line is largely unexplored, and it is not clear if White has enough for two pawns, but White certainly gets a persistent initiative. Most of the time, though, Black is not keen on accepting pawns after having originally declined the initial pawn sacrifice with 4...d5. 5...Nf6 6.e5 Ne4 is met by another pawn sacrifice; 7.0-0, which appears to be sound. 5...dxe4 6.Bxe4 Nf6 7.Bxc6+ bxc6 8.0-0 (a suggestion of Gary Lane) leads to an equal game, again with White having an isolated pawn on d4, but compensatory long-term piece activity. I can only find 58 games with 5.Bd3 in the database, with a decent score of 53% for White.
Not recommended, however, is 5.e5?, whereupon Black should grab the pawn on c3 with 5...dxc3 and if 6.Nxc3 then 6...d4 disrupts White's development.
Other ways of declining
4...Nf6 is another way of undermining White's e4-pawn, transposing into a line of the Ponziani Opening (1.e4 e5 2.Nf3 Nc6 3.c3 Nf6 4.d4 exd4) and if 5.Bd3?! then Black should take the pawn, 5...dxc3, and steer the game into lines where White's light-squared bishop would be far better placed on c4. So White must push the e-pawn: 5.e5.
Then Black can choose 5...Nd5, which leads to solid positions where White may have a slight edge after 6.cxd4. 6.Bb5 tends to come to the same thing but offers White some more gambit possibilities, e.g. 6...a6 7.Ba4 Nb6 8.Bb3, allowing 8...dxc3. Tim Harding has recommended this line for Black, but in the database White is scoring a very healthy 58% against it.
5...Ne4 is more counterattacking and double-edged, but more likely to concede a significant advantage to White. It is a decent practical try though, scoring 43% for Black. The main line sees White trying to "pin and win" Black's knight with 6.Qe2 f5 7.exf6 d5 8.Nbd2, whereupon 8...d3!? is probably Black's best way to mix things up and get a fair amount of counterplay, though with accurate play White is better.
4...Nge7 intends 5...d5. In the database, albeit from just 30 games, Black is scoring an impressive 63% with this, so it deserves significant consideration. In the stem game Tartakower-Reti, Berlin 1928, Tartakower found the right response, 5.Bc4 d5 6.exd5 Nxd5 7.0-0, preparing to attack Black down the e-file. Black's game is playable but I think White has slightly the better chances. Many of White's losses in this variation have been in low-level games where White appeared to be taken by surprise by Black's offbeat fourth move.
The "push past" move 4...d3 is playable but, as with many such "push past" lines against gambits, consigns Black to a rather passive position. White is slightly better due to a lasting space advantage, without having had to sacrifice anything, and has scored 58% in practice. Black can also consider 4...Qe7, counterattacking against e4, but this gives White a pleasant choice between the simple and strong 5.Bd3, and freely sacrificing the pawn with 5.cxd4 Qxe4+ 6.Be3. Other 4th moves for Black, although not completely disastrous, allow White to get a strong pawn centre for free with 5.cxd4, and White scores in excess of 70% in practice in these strong-pawn-centre lines.