So here is my wife's situation:
- she is from non-EU country (middle-east) and has a US travel visa valid till 2016 on her passport.
- she used her US visa to enter the US for business trips in the last years
- she has now a swiss e-passport (biometric) valid under VWP (visa waiver program)
- she plans to go to the US in September

Question: what passport should she use and declare to the ETSA system?

I had a look at the US Travel Site which answers the question:
I have dual citizenship. Which passport should I use to travel to the United States?

with

"All U.S. citizens, even dual citizens/nationals, must enter and depart the United States using his/her U.S. passport."

Doesn't help much

If someone has a dual EU and US citizenship, they can travel to the US with the US, not the EU, or to the EU using the EU Passport. shouldn't be a problem. Though with a US Visa it may be different, I would try and call the homeland security in the US beforehand, it isn't something to be messed with. I heard of an individual whom had a green card, stayed out of the US more than a year, and then returned. This person was apprehended at customs, had a choice, stand trial for visa violation or get deported, the decision this person made was be deported back to the EU, in hand cuffs, accompanied by a US Customs Agent, yes, totally unnecessary, but it happened.

Either would be fine, the Swiss is probably better, phone up the US Embassy in Bern and ask them.

Tom

There is a big difference of having a tourist/business visa in the passport for a couple years, and having a green card.

A green card holder in the US, is not allowed to abandon permanent residency. If a green card plans to leave the US for more then a year,
one has to obtain a "Re-Entry permit" prior to leaving the US, which is basically a white re-entry travel document. (Also used as a stateless
passport, for US permanent immigrants with no citizenship).

Based on the OP's description, I think that the Swiss passport is the "stronger" one, and will cause less headaches. She is considered a Swiss citizen, correct? Any western EU passport will be better than a non-EU passport.

Carry the other one for backup, but do not offer it as identification.

ETA. I think this is more applicable:

ESTA is only applicable if you are applying for admission under the visa waiver program (in this case, the Swiss passport is eligible for ESTA).

She can use either passport she chooses. However, if she's using the Swiss one, I'd recommend she be prepared for additional questions, as they will see that the last admission was on a different passport and it may raise suspicion for that particular officer.

The Department of State website didn't help you because it is geared to US Citizens. For non-citizen traveler information, you will want to check http://www.cbp.gov/xp/cgov/travel/

Crystal clear. It's exactly the answer I need. Thanks very much. I'll print it and ask my wife to have it close at hand when crossing the US border