Please note that this is not the same as a +1 influence on these cards. This is a penalty that also applies to in-faction cards as well. Check out the official announcement and the new Tournament Rules.
The affected runner cards:
This whole stuff smells like Prepaid Kate. The popular archetype (example) is being used by a lot of players. These decks show very little variance. This deck is tweaked as it can be and Kate is the most popular top 30% runner identity in the meta.
After the nerf, this archetype would lose 9 influence, which is devastating. You can drop the whole "prepaid" concept and go with other economy sources, Professional Contacts and Technical Writer seems nice. You can try other Kate variants, for example: Nexus Kate.
If you are sticking with the Prepaid VoicePADs you will have to cut cards hard. You can exchange some of the Clone Chips with Scavenge. This enables you to use one Cerberus "Lady" H1 less and also opens up play with D4v1d. You can even throw in a Test Run for good measure. Try taking some Lucky Find out which is sad, but times are hard. If your gamble is that fast advance will not be relevant, Clot can be left out as well.
The vast majority of Shapers were relying on Clone Chip heavily and using Cerberus "Lady" H1. I guess they can handle the situation they way I just outlined. Hayley Kaplan: Universal Scholar or Chaos Theory: Wünderkind can be next big thing if Kate falls. Kate still has a very powerful ability which is hard to compete against. Poor Rielle "Kit" Peddler will have a hard time with her original 10 influence limit, but I wager the players can make some stealth variation work.
This guy. Since an ID's influence cannot be reduced below 1, The Professor: Keeper of Knowledge is not affected by these new rules at all. I'm not sure that he becomes tournament-viable, but he has gained some power compared to the other identities.
All successful non-DLR Anarch decks were using Parasite and Clone Chip. Look at Faust Noise. Archetypes like RegAss Valencia, L4J Whizzard, RegAss Maxx even used Yog.0 as well. This is a massive hit. They will probably ditch most of the Clone Chips and rely more on Déjà Vu, Trope, Same Old Thing and Retrieval Run. Without Clone Chip, playing Parasite is less effective, but they will probably hold onto them. The "cutlery" events are a substitute option, also Immolation Script is nice for Noise. They will splash less Account Siphon and Desperado for sure. Maybe Quetzal: Free Spirit will become stronger as other identities weaken.
As for the Data Leak Reversal (DLR) decks, I'm puzzled. They are not using the cards on the list, so they receive no influence penalty. The fact that you can only have one Wireless Net Pavilion on the table is harsh. The key to DLR play was to put economic pressure on the corp, so it cannot trash key resources. Now this becomes much cheaper. Maybe DLR will fade or move toward Criminals.
They were kicked in the face less than the other factions, but sapping Desperado hurts. It might be the best console ever, so I understand that FFG tries to push players towards other Criminal consoles. Influence is quite tight in this faction, it usually pays for breakers. Perhaps Criminals will use their central breaker suite and Faust for remotes.
I'm having my doubts that this nerf will do Criminals any good. With less Clone Chip being played, breaker destruction will be a valid play style for the corporation. If this happens, Criminals are f*cked. All other factions have access to program recursion in some way, but they don't.
The affected corporation cards:
So if you look at your average NBN astrobiotics deck, this means a whopping 11 penalty in influence. Clearly Damon (current designer of Netrunner at FFG) had enough of this archetype. Somewhat comforting, Near-Earth Hub: Broadcast Center starts at the unreasonable 17 influence. You will probably keep the AstroScript Pilot Programs, the SanSan City Grids, with the option of using Explode-a-Palooza instead of NAPD Contract. The place where you will probably cut your losses is the ICE. NBN lacks good end-the-run ICE, so the chance that the runner will just steamroll you will be greater than before. You can try going with the tags using Data Raven, Gutenberg, Resistor, maybe Turnpike. Just be sure to punish the tags somehow. Psychographics is a fast-advance solution, Closed Accounts and The All-Seeing I can yield some painful in-faction moments.
What about the butcher shop? Still possible, but the influence becomes really scarce. These decks used to have a fast-advance "plan B" if the murder fails which was also a nice pressure option for the runner. Now this is almost impossible. Maybe if you are willing to sacrifice Traffic Accidents. Bottom line is this archetype also loses power.
The other NBN identities will face the same problem. They either sacrifice the fast-advance option or the good end-the-run ICE. The default agenda composition is stirred. If NBN players lean toward the glacier style, AstroScript Pilot Program will be left out and we will see Global Food Initiative more often. Project Beale+Psychographics, Quantum Predictive Model and maybe even Market Research will be played if you can tag the runner effectively. Glaciers need a lot of money for big ICE. The new identities like Spark Agency: Worldswide Reach and New Angeles Sol: Your News (with Paywall Implementation) might be able to pull that off while protecting their servers with Tollbooth, Archangel and Ash 2X3ZB9CY.
Foodcoats is in bad shape also, Eli 1.0, Architect and NAPD Contract are nerfed. I find Architect the easiest to rid, the Ichies, Rototurret and Assassin can take its place. Eli 1.0 is such a wonderful taxer, you either still keep it, or you can replace it with Viper, Enigma, perhaps Markus 1.0. Dropping NAPD Contract would leave a gaping hole, you might fill some with Project Vitruvius. Reducing Caprice Nisei to regain some influence is an option, but a grievous one.
NEXT decks might be a good thing again, provided that people will play less Parasite and Clone Chip. In that case, Marcus Batty and NEXT Gold become heavy hitters. I'm skeptic about other HB identities.
Jinteki has some awesome multi-sub ICE like Komainu and Tsurugi. The problem was that they have low strength and were vulnerable to Parasite. Now they will become more playable and thus offer a taxing solution against the much abused Faust.
Replicating Perfection decks were having problems playing against Noise: Hacker Extraordinaire, this becomes less relevant. It will probably become a top-tier archetype again. Jinteki: Personal Evolution (example) and Industrial Genomics: Growing Solutions (example) can be successful as well. These decks need to cut some Eli 1.0s and they should be fine.
Weyland is the king of barriers, less Cerberus "Lady" H1 and Parasite will help it. This is the faction is least affected by the NAPD Most Wanted list, only NAPD Contract comes into mind. I'm sure players will manage to squeeze something else in or find the required influence. The changes overall will surely make Weyland more relevant than before.
Edit: Check out the Winning Agenda podcast interview with Damon Stone here.