The Crucible expansion has a lot of content in it. There are small changes and large changes, and both kinds can have small impacts or large impacts. Accurately determining the fallout of all of these at once is nearly impossible, but some guesses can be made by people with a long view of the game and the desire to increase blog post counts.
Instead of talking about how each change will impact the game separately, I will try and look at how each area of the game will be impacted at least in one regard, if not a couple.
With the proposed changes that no insurance will be paid out for death by CONCORD and that autopilot will actually dock you in your destination station, one can see that suicide ganking is the direct target of these initiatives. On the other hand, a large majority of ganks occurred by people who didn't really depend on the insurance money to be profitable and the ganks typically took place against traveling targets near the trade hubs and routes. While this might mean that suiciders are more selective and make greater use of ship scanners, I doubt we will see a large drop in the number of suicide ganks. In fact, the introduction of the new pocket battleship battlecruisers may mean an increase in ganks as the perpetrators can use cheaper hulls with battleship weapon alpha strikes.
Also, I predict an increase in suicide ganking of autopiloting pods due to the inclusion of implants on kill mails in order for some to try and increase kill board efficiency.
The player owned customs offices could introduce some intriguing game play in low sec. Currently a number of players harvest resources of low sec planets on the sly, slipping in with blockade runners to pick up there goods and moving them to other planets or to be used elsewhere. With the public offices gone players will be forced to either abandon their networks or try to deploy their own offices... that can be attacked and destroyed. Most will be unwilling to accept that risk (realizing that they would be unable to protect their assets) so I expect low sec planetary interaction to shrink and stagnate in the short term. Longer term we might see alliances step in with the wherewithal to setup and defend the more rare low sec planets with private customs offices (or perhaps public with fees to generate income).
The new battlecruisers will not have a huge impact in the fighting seen in low sec since a lot of low sec fighting involves smaller hull sizes where the battleship caliber weapons will not be needed. The changes to hybrid weapons and hybrid ships, however, could have a large impact. Expect the Dramiel to fall out of favour as ships like the Daredevil come to the fore, as well as Gallente blaster ships in general.
Also, with destroyers getting a buff (rate of fire penalty gone, more hit points, lower sig, more cap) you will definitely see more destroyers in all fields of battle, but especially low sec areas where hostile frigates can be found more often, especially in area of faction warfare.
I'm going to go out on a limb here and say that the new year will see a lot of alliances lose their space and warfare to burn across all areas of low sec. Alliances that once were able to leverage large super capital fleets to force their enemies to step down will find that their super carriers are vulnerable to fleets of the new pocket battleships (i.e. fighter bombers cannot hit them while they do battleship damage for a cheap price) while titan doomsdays will not clear away the enemy logi and leadership vessels. In addition, the new aggression rules upon logging out means that that risk of using your multi-tens-of-billions ships is a lot higher. As an upshot of this, along with the improvements to Dreadnoughts and addition of tech II capital mods, expect to see a lot more Dreadnought and Carrier fleets as the backbone of sov warfare.
But wait! There's more...
Alliances in null sec will make a concentrated effort to erect custom offices in their territory to keep the POS fuel products rolling in as POS fuel prices increase due to small operation and ninja PI industrialists close up shop in low sec and null sec. These structures will be favourite targets to attack for smaller hostile fleets as they will lack any automated protection and they might stir the locals to form a defense fleet, thus getting a "gud fite". Even if that fails, there is a chance for a fight when the reinforcement timer comes out. So expect a lot more action around customs offices in null sec.
But WAIT! THERE IS STILL MORE!
Perhaps the biggest change of Crucible to impact null sec will be time dilation which promises more even footing in the lag battlespace for both sides of a fight. This in turn will encourage more people to come out for big fights as they feel they stand a chance of actually doing and seeing something, even at a snail's pace. Therefore I think we'll see the first 4000 pilot battle in a single system in early 2012.
The new battlecruisers will play a fairly big part in Crucible's impact on wormhole space. Since they sport battleship firepower on a battlecruiser hull, it will offer more options for attacking enemy structures through smaller wormholes or wormholes with mass restrictions. Adding into that equation the fact that customs offices will be owned most likely by the wormhole system inhabitants, it gives another vector to attack neighbours. Therefore I predict more structure shooting and corporate evictions in class 1 and 2 wormholes.
Also, wormholers will be deliriously happy over corporate bookmarks.
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That is my main predictions for life in New Eden post-Crucible. If you have your own crystal ball readings, please post them in the comments!