In answer to this question we must say “It depends.”
Some folks think that Catholic acceptance of any Protestant Baptism at all is a Vatican II novelty. This is not the case. Here is what the Catechism of the Council of Trent says:
Those who may administer Baptism in case of necessity, but without its solemn ceremonies, hold the last place; and in this class are included all, even the laity, men and women, to whatever sect they may belong. This office extends in case of necessity, even to Jews, infidels and heretics, provided, however, they intend to do what the Catholic Church does in that act of her ministry. These things were established by many decrees of the ancient Fathers and Councils; and the holy Council of Trent denounces anathema against those who dare to say, that Baptism, even when administered by heretics, in the name of the Father, and of the Son, and of the Holy Ghost, with the intention of doing what the Church does, is not true Baptism. [source; emphasis added]
Some might object that Protestant intentions in baptism are not Catholic ones, but that is not how the Church sees things. As Pope Leo XIII said in 1896:
The Church does not judge about the mind and intention, in so far as it is something by its nature internal; but in so far as it is manifested externally she is bound to judge concerning it. A person who has correctly and seriously used the requisite matter and form to effect and confer a sacrament is presumed for that very reason to have intended to do (intendisse) what the Church does. On this principle rests the doctrine that a Sacrament is truly conferred by the ministry of one who is a heretic or unbaptized, provided the Catholic rite be employed. [Apostolicae Curae 33; emphasis added]
The Church does not judge the mind, just because no man can know another’s heart. So instead it is actions that are considered, and if the actions in a Baptism include the correct matter and form, then the presumption is that the Baptism is valid. So the next and obvious question is: what is the matter and form of Baptism? The Catechism of the Council of Trent informs us:
Now since we said above, when treating of the Sacraments in general, that every Sacrament consists of matter and form, it is therefore necessary that pastors point out what constitutes each of these in Baptism. The matter, then, or element of this Sacrament, is any sort of natural water, which is simply and without qualification commonly called water, be it sea water, river water, water from a pond, well or fountain.
Pastors, therefore, should teach, in clear, unambiguous language, intelligible to every capacity, that the true and essential form of Baptism is: I baptise thee in the name of the Father, and of the Son, and of the Holy Ghost. For so it was delivered by our Lord and Saviour when, as we read in St. Matthew He gave to His Apostles the command: Going,… teach ye all nations: baptising them in the name of the Father, and of the Son, and of the Holy Ghost. [source]
So we see that if Protestants baptize with water using the correct words (“I baptize you in the name of the Father and of the Son and of the Holy Spirit”), the baptism is considered to be valid, and this is no “novelty” of Vatican II. It’s reasonable to be uncertain in specific cases, until we know whether the correct form of words was used or not. But for the vast majority of Protestants, this is just not an issue. Baptists, Methodists, Anglicans, Presbyterians, Lutherans all use the correct formula.