It’s mostly depends on the characters involved, the time/health/meter situation, and your understanding of the opponent’s habits. But basically there are three main approaches to corner combat from the perspective of the aggressor:
- Apply pressure
- Maintain position
- Ignore the opponent’s situation and take advantage of the real estate behind you
In SF2, zoning characters can perform option 2 using fireball traps (it can also be done in SFIV but it’s harder). An opponent cannot take chip damage forever (constantly chipping at the opponent’s health is also form of pressure), and eventually they will have to guess their way out by perform a risky escape move, or rely on you messing up. Characters without fireballs can still perform a variation of this if they have a good anti-air, and a move that can safely chip the opponent while allowing them to reset the corner game eg. Fei Long or Boxer in SSFIV. Fei can slowly chip at you with rekkas and force you to attempt a risky counter or escape. Boxer and chip you with safe dash punches and punish all your jumps etc. Note that when these characters have the life lead, there’s no reason for them to chip damage pressure. The onus is on the opponent to try to escape the corner.
Option 1 is usually employed by characters that have good mobility, get good frame advantage off safe attacks, and/or have strong mixups. Also, when time is running down an aggressor who is behind on life is more likely to take risks by attacking (there is a mindgame here of course). Dictator (M. Bison) is a good example of a character that has a really good corner pressure game. He doesn’t do as well and Fei or Boxer at waging wars of attrition because his anti-air game is weaker, so it usually in his best interest to apply scissor-kick pressure. Even though they are good zoners, Juri and Guile and also apply strong corner rushdown pressure against certain characters because of the frame advantage they get off the projectiles and normals. Juri has the added advantage of having a more effective overhead, and not requiring charge for her pressure. Vanilla Sagat has a very good corner game. He can back off and zone you out with a very strong fireball/anti-air game, but he could also keep an opponent locked down using safe Tiger Knee pressure.
Option 3 usually only comes into play when time is running down and you want to put as much distance between yourself and the opponent as possible in order to score a win by Time Out.
For an example of how meter can affect the corner game, consider the situation between Ryu and a cornered Blanka in AE2012. When Blanka has meter, he can use EX Rainbow Ball to escape rushdown corner pressure. Ryu can use an option select to defeat this, but this will lose to Blanka’s EX Vertical Ball. Ryu can try to bait the EX V. Ball and punish, but he loses momentum and risks Blanka escaping by doing a normal into EX Rainbow, or perhaps hopping right through him. In this scenario, it’s better for Ryu to maintain a position where he can comfortably anti-air, and punish EX Rainbow with cr.hp. However, when Blanka does not have meter, Ryu can safely rush him down and Blanka’s options are essentially limited to blocking (or risking an Ultra combo).
Remember that even though a cornered the opponent is usually at a big disadvantage, if there’s no pressure on them (in terms of life deficit), they have no reason to try to escape - you will need to give them a reason to try to escape the corner. Similarly, an opponent whose character has an easy escape from the corner (eg. teleport) would not usually be afraid of getting rushed down. And in some situations, an opponent might be completely comfortable by playing from the corner. This usually happens if they have a life lead and your character has a tough time getting in on them.
As for what triggers “panic mode”, it’s usually a combination of a life deficit and clock that is running down. Good players don’t usually panic easily, but in general most players in that situation will tend to take more/bigger risks. Regarding opponents with a good Anti-Air, you should avoid jumping at opponents in general unless it’s a punish situation. And when they are cornered, there’s even less of a reason for you to jump at them unless it’s a knockdown setup or mixup.