What is the Baptist response to 1 Corinthians 7:14 if the children are believers but the parents are not?

Firstly, please remember there is no such thing as "Baptist theology" except that a person should be baptised upon their own profession of faith in Christ. Even that definition might be stretching Baptist theology too far because some who call themselves "Baptist" are so liberal that they might not want to include "in Christ" in the definition. The baptist persuasion allows for the independence of the local church, and therefore every local church could have its own theology. So most are Armenian, some are Calvinist, some are evangelical and some are liberal, etc. And no sole authority exists to decide what is Baptist and what is not.

Secondly, some well-known Baptist ministers have taught that infants who die before they reach a certain age are included in the benefits of the work of Christ and are eternally saved even though they never expressed faith on Christ. And they would hold this to be true not just for the children of a believer but of all children everywhere. See the sermon entitled "Infant Salvation" by leading Victorian-era London Baptist pastor Charles Spurgeon here https://www.spurgeon.org/resource-library/sermons/infant-salvation/#flipbook/

Thirdly, this passage of 1 Cor 7:12-16 gains added importance when we remember that in the Old Testament Nehemiah remonstrated with the Jews who had married or had allowed their sons to marry non-Jewish women (Neh 13:23-28); and Ezra the scribe urged Jewish men to divorce their non-Jewish wives (Ezra chapters 9 and 10, eg Ezra 10:19).

In the light of those OT passages should NT believers divorce their unbelieving spouses? No, says Paul. And that is the main purpose of this section: it has nothing to do with the salvation of the children of believers; to argue that is to import into the passage something foreign to the passage's purpose.

Finally Baptist pastor Dr Peter Masters argues that verse 1 Cor 7:14 means something like this:-

“For the unbelieving husband is set apart for the wife, and the unbelieving wife is set apart for the husband. Otherwise your children would be illegitimate, but as it is, they are legitimate.”

The idea is that the spouse is not made holy, sanctified in any religious sense, but that God has “set them apart” to be the husband/wife of the believer. “Sanctified” has the idea of “setting apart” for holy use: God has set the unbelieving spouse apart for the believer, and because God thus legitimises the marriage therefore the children are legitimate also.

A believing parent has no Biblical warrant for believing there is any stigma attached to the children because one of their parents is an unbeliever.

The main theme of this passage is essentially that believers are not to try to divorce their unbelieving spouses purely on the grounds they are unbelievers.

Let there be peace in, and submission to, the providence of God.

We should not be even attempting to understand what "the children are holy" means outside the context of the primary intention of the passage.

This chapter is primarily answering questions asked of Paul by the church in Corinth.

Now concerning the matters about which you wrote:... 1 Cor 7:1

Some of those questions had to do with the state of marriage once the new birth enters in to the equation;

Verses 1-6 treat the matter of conjugal responsibilities within an existing marriage (permanent marital continence is strongly discouraged) .

verses 7-9 treat marrying as a protection against carnal desire for those eligible (the ability to remain celibate is from God, not an act of human will)

verses 10-11 prohibit divorce (and remarriage) and I think the following verses indicate that these two are regarding marriage between believing spouses.

verses 12-16 treat the matter of this new faith as a potential division within an existing marriage and the verse in question mentioning children is in the context of Paul responding to questions regarding the marriage dynamic where one spouse has come to faith and the other has not. It is not explicitly given that the state of children within this sort of a marriage was a part of the original questions, however it is reasonable to assume that it may be the case. It is also reasonable to read the particular phrase,

Otherwise your children would be unclean, but as it is, they are holy. (v. 14b)

in context, as a parenthetical explanation of the principle being addressed in the beginning of the same verse.

For the unbelieving husband is made holy because of his wife, and the unbelieving wife is made holy because of her husband. (v. 14a)

In other words, "If the principle of the unbelieving spouse being made holy by the believing spouse were not true then your children would not be holy but, as it is, they are."

This doesn't show us the difference that Paul intends to impart by using the terms 'unclean' and 'holy' but it does serve to illustrate that there is a participation in the 'holiness' (whatever that is) of the one by another. But then in v. 15-16 it becomes clear that an unbelieving spouse being 'made holy' by the believing spouse is not the same as being 'saved':

But if the unbelieving partner separates, let it be so. In such cases the brother or sister is not enslaved. God has called you to peace. For how do you know, wife, whether you will save your husband? Or how do you know, husband, whether you will save your wife? (v. 15-16)

Here we have the believer/unbeliever couple and the believing spouse (who is currently making the unbelieving spouse holy in the marriage) may let the unbelieving spouse go if they wish to leave BECAUSE the unbeliever's final state as regards salvation and the believer's role in that salvation is unknown.

If the holiness imparted to the unbelieving spouse is not equated with salvation then the holiness imparted to the children according to the same principle should not be either. Therefore many of the attempts to explain what Paul means here are erroneous because, in their reasoning, holiness is confused with salvation.

So then, what meaning do these words, 'unclean' and 'holy', try to impart in this context if not salvation? The Hebrew word ko-desh (Holy) carries the meaning of being separated or set apart. We read into it a hinting of moral uprightness (which uprightness certainly should be a part of being set apart) but this is not intrinsically necessary in the definition; a plate, an animal, or an article of clothing can be Holy unto the Lord, that is to say dedicated for His use.

In reference to God it means that there is absolutely no other like Him; He is creator of all and utterly unlike His creation. In reference to believers it carries the meaning of being called out from one thing unto another thing and very often the transition is from uselessness in the Kingdom to usefulness. This all hearkens back to the system of the temple and the priests where certain items (and individuals) were acceptable for use in service to God. Some of those things were only ever to be used for that one purpose (they were always clean for use). Others could be used if certain cleansing rituals were undergone (they were available and could be cleansed). Still others were not allowed (they could not be cleansed).

Since the verse in question (1 Cor. 7:14) does not equate holiness with salvation and the terms holy and unclean can be shown to indicate availability for divine usage, Paul's point regarding these children might be that as long as there is at least one believer in the household the others in the household are thereby made available for cleansing. They are made holy in the sense that they are made available (to be cleansed and used) by their proximity in relationship to the believer (who is cleansed and useful) as opposed to being unclean (and not cleanable).

The final verse (16) makes sense in this light because the believing spouse is encouraged neither to expel nor retain the unbelieving spouse but rather to allow staying or going as they desire BECAUSE the believer can not know whether or not their relationship will result in salvation.

To further the idea that "holy" can apply to children prior to their coming to personal belief we have:

And when the time came for their purification according to the Law of Moses, they brought him up to Jerusalem to present him to the Lord (as it is written in the Law of the Lord, “Every male who first opens the womb shall be called holy to the Lord”) - Luke 2:22-23 

Holy here is the same as holy in 1 Corinthians 7 and carries the meaning of separated entirely for the use of the Lord. Therefore, children in a marriage where only one spouse is a believer are thus holy (separated) unto the Lord. They will still need to be born again by grace through faith.

What does not appear to be taken into your question is the age of consent. That is a strict belief in the Baptist Denomination. While a child is under the age of consent they are covered by the Grace of God (or in other words they are not judged by their not accepting Jesus as their Savior. They are accepted as were Adam and Eve prior to the fall).

Later when they are of age it is taken into consideration whether or not their parents of someone else have presented the path of Salvation with them.

Matthew 18:10 KJV Take heed that ye despise not one of these little ones; for I say unto you, That in heaven their angels do always behold the face of my Father which is in heaven.

Here we see that prior to the Crucifixion these little ones received special consideration in the face of God.

Matthew 24:14 KJV  And this gospel of the kingdom shall be preached in all the world for a witness unto all nations; and then shall the end come.

Here we see that judgment of the World will not come until all of the World has seen the opportunity to accept Christ as their Savior. This requirement is that those capable of making that decision must elect Salvation. We also believe that God is love and would not condemn anyone who for any reason was incapable of making that decision, to the outer darkness (or what is better known as the lake of fire).

There is of course much more included in the Baptist Denomination about eligibility for Salvation than this, but I hope this gives you the basic understanding of the Baptist concept of your question.