In answering that question I wrote the following, it is not meant to be an exhaustive answer, but it does focus in on some of the benefits children gain from being members of the visible church:

While there probably isn't much visible difference if you compare the children raised in well-ordered Presbyterian homes to children raised in well-ordered Reformed Baptist homes, there is a substantial difference in their relation to the church. The children of Presbyterian churches are actually members of the visible church. Thus they are not treated in the same way as unbelievers or "Canaanites," but are regarded as members of the covenant community and heirs of the promises. They are on the roll of the church, and under its oversight and pastoral care. In most respects therefore, they are like the children of Israel; they are set apart and federally holy and have the sign and seal of the covenant applied to them as members of the church. Therefore, unlike unbelievers, their parents can pray for and with them, expect them to worship and to tithe as soon as they are able. I have no problem them calling them my little "brothers" and "sisters." However, as was the case with Israel, we accept that the bounds of the invisible church and the visible church are not exactly coterminous, and that some of the children of the church, like Hophni and Phineas, will ultimately prove to be tares rather than wheat when they come to adult years. Both the word and experience teach us that not all the children of the church are elect, and those reprobate children will never "make good their baptism" and obey the gospel command to close with Christ by faith alone. In the final assize, therefore, their baptism and the greater benefits they had than the children of pagans will actually stand against them.

Also, in exhorting the children of the church from the pulpit one would not speak to them as one would an unbeliever outside the church, rather one would speak of the blessings and promises given to them, the responsibilities incumbent upon them and warn them of the grave danger of not making the God of their fathers their God by their own personal living faith in their Redeemer. As my favorite quote from Guthrie puts it so well:

"Believing on Christ must be personal; a man himself and in his own proper person must close with Christ Jesus—‘The just shall live by his faith.’ (Hab. 2:4.) This says, that it will not suffice for a man’s safety and relief, that he is in covenant with God as a born member of the visible church, by virtue of the parent’s subjection to God’s ordinances: neither will it suffice that the person had the initiating seal of baptism added, and that he then virtually engaged to seek salvation by Christ’s blood, as all infants do: neither does it suffice that men are come of believing parents; their faith will not instate their children into a right to the spiritual blessings of the covenant; neither will it suffice that parents did, in some respects, engage for their children, and give them away unto God: all these things do not avail. The children of the kingdom and of godly predecessors are cast out. Unless a man in his own person have faith in Christ Jesus, and with his own heart approve and acquiesce in that device of saving sinners, he cannot be saved. I grant, this faith is given unto him by Christ; but certain it is, that it must be personal."

I should note that I do not regard it as safe or right to preach as though it is absolutely certain that all the adult hearers in the church are necessarily regenerate, but rather to simply preach and apply the gospel calling upon all men to repent and believe. Hearing the preaching of the gospel and the call to faith will certainly do no harm to the regenerate in any event!

On a personal note, as someone raised outside the visible church and not baptized until I made a profession of faith in adult years, I remember envying the covenant kids in seminary. Not only didn't they have a host of wasted years and memories they'd rather not be walking around with, they had been raised under the instruction of scripture and the nurture of the church. I was always amazed when they treated their inheritance as nothing special and showed more interest in the wasteland I'd come out of.