It's -25% for crossing a river, and -50% for attacking from a transport.
It's -75% if you "cross" a river from a transport (i.e. attack diagonally over where a river would be if it extended into the sea).
Marines/Navy SEALs don't suffer this penalty, but they also aren't eligible for the City Raider promotions.
For a modern military, I like to stick with Modern Armor. No, it can't ever get Amphibious, but it does get City Raider.
Some comparisons between Marines and Modern Armors when attacking from the sea, using the best promotions:
Level 1:
Marine = 24
MA = 20
Level 2:
Marine = 26.4 (Combat I)
MA = 24 (City Raider I)
Level 3:
Marine = 28.8 (C II)
MA = 29 (CR II)
Level 4:
Marine = 31.2 (C III)
MA = 35 (CR III)
Level 5:
Marine = 33.6 (C IV)
MA = 37 (CR III, C I)
Level 6:
Marine = 36 (C V)
MA = 39 (CR III, C II)
Level 7 (versus gunpowder):
Marine = 42 (C V, Pinch)
MA = 46 (CR III, C II, Pinch) or 43 (CR III, C III)
I've modified the attacker strength for simplicity. The actual calculations are a bit more complex, with all upgrades save for Combat modifying the defender's strength (i.e. a bare MA will have a strength of 40, but the strength of the defender will go up because of the amphibious assault).
As you can see, once you get to level 3, the Marine starts to fall behind, and continues to do so. The amount of damage each attack round does depends on attacker strength, too, so the MA has an implicit advantage there. The MA also has 1 first strike, though Navy SEALs have 1-2 first strikes.
If you get past level 7, you can get Ambush for the Marine (+25% versus armor). After that, you could push for Guerilla III (Warlords only), which gives +25% attacking hills. But I'd take advantage of the fact that the Marine gets defense bonuses, and give him the City Garrison upgrades. A Marine with all three City Garrison upgrades won't be quite as good a city attacker as a MA, but it will be slightly better at defense (and only get better if it fortifies, or the city is on a hill).
For MA, fill up the Combat upgrades, then add Ambush. In vanilla Civ4, I'd go with Barrage after that, but in Warlords, collateral damage is nerfed, so you don't get any real benefits until you get Barrage III (Mechs are immune to Barrage I & II if they're already at 50% hp). Better to pursue Drill, to get more first strikes (less vulnerable to long runs of bad random numbers).
The reason why you want to attack from sea in the first place, of course, is that your attacking units are immune to damage up until you use them. If you land first, you will get hammered by jets, bombers, and artillery.
)
) with 'frigate', loaded up with with 'cannon'. And discovered it didn't work.
Cheap and effective. Add a mech infantry onto the transport to defend the city afterwards, and airlift a SAM inf.
