My understanding is that they are neither required nor forbidden to sing (or make any gestures, etc.) The EF, Fr. Z often says, has no congregational rubrics. My advice would be to go on local custom.

In his letter accompanying "Summorum Pontificum," Benedict makes suggests that some interplay between the two forms is desirable:

"the two Forms of the usage of the Roman Rite can be mutually enriching: new Saints and some of the new Prefaces can and should be inserted in the old Missal. The "Ecclesia Dei" Commission, in contact with various bodies devoted to the usus antiquior, will study the practical possibilities in this regard. The celebration of the Mass according to the Missal of Paul VI will be able to demonstrate, more powerfully than has been the case hitherto, the sacrality which attracts many people to the former usage. The most sure guarantee that the Missal of Paul VI can unite parish communities and be loved by them consists in its being celebrated with great reverence in harmony with the liturgical directives. This will bring out the spiritual richness and the theological depth of this Missal."

There seems to be a trend in celebrating the 1963 Missal as if it were 1963. This is, following not only the rubrics but also the devotions and customs and musical trends that were popular at the time. I don't mean to pass judgement on those who have a sentimental attachment to the EF Mass, but I do think care should be taken that the Rite isn't seen by those unfamiliar with it as a mere museum piece. Some developments of the of the past 40 years might enrich the traditional celebration.

To respond to the posted question, the full, conscious, and active participation of the assembly still applies to both forms. We of course know that the participation is both internal and external. If Vatican II has been successful at all in improving people's understanding of the Mass and ability to participate in it, then they should be able to participate in the EF more fully than was previously possible. Now I don't work in the EF, but if I did I think I would concentrate on having people sing those things that I would expect them to sing in an OF Mass, viz Kyrie eleison (in alternation with the choir), Agnus, Sanctus, and Gloria. I don't see any reason why a congregation couldn't sing the "Ite" as well, though this is rarely done in the OF.

For other parts of the Mass I would focus on people's internal participation in full chant or polyphonic propers. Only then would I work on supplemental hymns in Latin or, where permitted, the vernacular.

I'm probably going way beyond the original question about responses, but I addressed the parts of the Mass that a congregation might need direction in order to participate in. I feel that if a congregation has, say, the PBC to follow along during Mass, they can choose which other parts they would like to sing and can learn them on their own without being forced or asked to.

Congregational participation at EF Masses seems to be very much local custom. And that custom can change. One Mass I occasionally attended had fairly good congregational participation on the ordinary and responses. Two years later, they had fallen silent. A year after that, the speaking/singing is picking up again. And I have no explanation.

At the parishes I go to, it's "DOMinmumblemumblemumble.... DOMinmumblemumblemumble.... DOMinmumblemumblemumble...." I try to join in but that's some tough Latin to rattle off! Drives home the "non sum dignus"!