I was born in Yemen, but to British parents and never had Yemeni citizenship. Can I travel to the USA under the Trump ban?

CBP put out a series of questions and answers on February 1, which said that the ban on entry and visas will only be enforced based on the passport that is presented by the traveler. Even if you had dual nationality with one of the 7 countries, it wouldn't matter as long as you present the passport of another country:

Does this Executive Order apply to dual nationals of the seven countries who want to enter the United States? If they apply for entry based on their citizenship from one of the countries NOT on the list, will they be allowed entry?

Travelers are being processed and, when eligible, admitted according to the travel document they present.

Can a dual national traveling with a passport from an unrestricted country travel to the United States?

Dual nationals with a valid immigrant or nonimmigrant visa in a passport issued by any country not restricted under the Executive Order will be permitted to apply for admission to the United States.

Can a dual national who holds nationality with a restricted country and is currently overseas, apply for an immigrant or nonimmigrant visa to the United States?

Department of State’s Posts are allowed to process visa applications and issue nonimmigrant and immigrant visas to otherwise eligible visa applicants who apply with a passport from an unrestricted country, even if they hold dual nationality from a restricted country. Please contact the Department of State with any questions related to the issuance of visas.

We've had a similar question recently about traveling to the US as a child of an Iranian citizen. It was a ban enacted by the Obama administration, but it is directly referenced by the Trump ban (that's where the list of 7 countries comes from), so I assume the interpretation should be the same. Unfortunately there doesn't seem to be an official guideline about people in your situation, but one data point mentions that:

I ended up going to the US without a visa and was allowed through with no problems. He didn't ask any questions about my nationality. An uncle I was traveling with, who is a British citizen but was born in Iran, got a visa and when he was going through customs and immigration was asked why he got a visa when he didn't need one! The officer said he would have been fine with just an approved ESTA...

So I assume you should be okay to enter, but it's also possible that the airline would err on the side of caution and deny you boarding. Overall I'd recommend waiting for a couple of months to see how things play out - at the very least there would be enough data points to answer your specific question.

According to a statement by the Foreign Office, only people travelling from those countries are targeted by the order. Even dual nationals will be allowed into the US. It appears safe to deduce from this, that a British national with a less strong link to Yemen will be allowed in the country:

The Foreign Secretary Boris Johnson has today held conversations with the US Government and as a result we can clarify that:

• The Presidential executive order only applies to individuals travelling from one of the seven named countries.

• If you are travelling to the US from anywhere other than one of those countries (for instance, the UK) the executive order does not apply to you and you will experience no extra checks regardless of your nationality or your place of birth.

• If you are a UK national who happens to be travelling from one of those countries to the US, then the order does not apply to you – even if you were born in one of those countries.

• If you are a dual citizen of one of those countries travelling to the US from OUTSIDE those countries then the order does not apply to you.

The only dual nationals who might have extra checks are those coming from one of the seven countries themselves – for example a UK-Libya dual national coming from Libya to the US.

The US has reaffirmed its strong commitment to the expeditious processing of all travellers from the United Kingdom.

(emphasis by me)

In how far this is known to border guards is another can of worms, to which I have no answer. At the moment I don't have an official US source stating the same information.

The US embassy in the UK, unfortunately, released a statement

Per U.S. Presidential Executive Order signed on January 27, 2017, visa issuance to aliens from the countries of Iraq, Iran, Libya, Somalia, Sudan, Syria and Yemen has been suspended effective immediately until further notification.

If you are a national, or dual national, of one of these countries, please do not schedule a visa appointment or pay any visa fees at this time.

If you already have an appointment scheduled, please DO NOT ATTEND your appointment as we will not be able to proceed with your visa interview. Please note that certain travel for official governmental purposes, related to official business at or on behalf of designated international organizations, on behalf of the North Atlantic Treaty Organization, or by certain officials is not subject to this suspension.

They do not state that dual nationals are banned (only that you will not get a visa at this time), which makes me think they put the unclear cases on hold awaiting clarification at the moment. They also make no statement about people born in those countries, which is your case.