A promise in Acts 16:31?

I think that the same basis requirements for salvation apply to everyone, as far as they are capable of complying. When Christ said "Be ye therefore perfect, even as your Father which is in heaven is perfect." (Matt. 5:48), He expected all those that chose to follow Him to obey to the best os their ability. But He also knew that everyone would fall short of that goal to some degree, each individual arriving to his own level. Then Christ, through His grace would make up for the rest, to the degree that each of us have qualified. Consider the following:

In Paul's epistle to the Hebrews, he gave a very simple answer to what is required for us to be saved. He said "And being made perfect, he became the author of eternal salvation unto all them that obey him;" (Hebrews 5:9) In other words, Christ only saves those that obey Him!

Assuming that by salvation you mean going to live in the kingdom of Heaven, Christ says very clearly, "Not every one that saith unto me, Lord, Lord, shall enter into the kingdom of heaven; but he that doeth the will of my Father which is in heaven." (Matt. 7:21) In other words, to obey Christ and gain salvation, we must do the will of the Father! Otherwise, He goes on to say that even though we might believe and act in His name, if what we do is not according to the will of His Father, He will not recognize us. (v. 22) More than that, He will condemn us for doing so. (v. 23)

He goes on to explain the will of His Father, saying, "Therefore whosoever heareth these sayings of mine, and doeth them, I will liken him unto a wise man, which built his house upon a rock: And the rain descended, and the floods came, and the winds blew, and beat upon that house; and it fell not: for it was founded upon a rock." (v. 25, 26) In other words, it is the will of the Father that we learn and obey all that Christ says. According to His words, this is required to enter into the kingdom of heaven.

Obviously then, to believe in Him is not enough! We must learn and do. He made this clear when speaking to some Jews that believed in Him:"Then said Jesus to those Jews which believed on him, If ye continue in my word, then are ye my disciples indeed; And ye shall know the truth, and the truth shall make you free."(John 8:31-32) In other words, again to obey Him, we must learn and do all that He said, to gain an accurate understanding of the truth that we might become free [of deception].

Regarding the importance of learning the truth in order to be saved, our Saviour also said,"And this is life eternal, that they might know thee the only true God, and Jesus Christ, whom thou hast sent." (John 17:3) In other words, gaining salvation or "life eternal" requires accurate knowledge of His Father and of Him. That is why we must continue studying His word under the guidance of the Spirit of Truth!

With these things in mind, then, our Saviour was probably able to make this promise to that household because He could see into their hearts and already knew that they were among those who would apply themselves to learn and live His teachings sufficiently to merit His Grace unto salvation.

To answer, it is necessary to analyze how the text is written, and to apply principles of theological interpretation. I will provide an answer which I think most Baptists or Evangelicals would find acceptable.

What was promised, and to who?

Paul is talking here to a specific jailer about salvation. He offers salvation -- the opportunity to be a member of Christ's kingdom -- on only one condition, that the jailer "believe". From many passages in the Bible, we can be confident that the belief is in Jesus, and that complete trust in Jesus is necessary.

Paul promises that salvation is available not just to the jailer, but also to his family. On what terms? Clearly, the jailer's belief is required. We could assume that the family's salvation is then automatic. But it seems likely that it requires them to believe also, and probably implies that their belief either is guaranteed to be the result of the jailer's belief and Christian life (if Paul is predicting the future as a prophet) or is the probable result of the jailer's belief and Christian life.

From Barnes Commentary:

That is, the same salvation is equally adapted to, and offered to your family. It does not mean that his family would be saved simply by his believing, but that the offers had reference to them as well as to himself; that they might be saved as well as he.

While the need for family members to believe also may be merely implied, Susan (Thanks!) pointed out in comments that the Greek can be interpreted as indicating that the condition of belief is intended to apply explicitly to both the jailer and his family members.

How does that apply to me?

We are definitely not promised that our families will be saved independently of believing, or even that they will necessarily believe because of our belief. Nowhere is it stated that others in similar situations will get similar results. What it does strongly suggest is that the incident was included in the Bible as an example of how Christianity was and can be spread. Each of us should have confidence that God desires the salvation of our relatives as much as we do, and confidently follow the leading of the Holy Spirit as we demonstrate the love and peace that comes with a position in Christ's kingdom during our daily walk with the Lord.

Again from Barnes Commentary:

It may be implied here that the faith of a father may be expected to be the means of the salvation of his family. It often is so in fact; but the direct meaning is, that salvation was offered to his family as well as himself, implying that if they believed they should also be saved.

Ezekiel 14:20 states:

even though Noah, Daniel, and Job were in it, as I live,” says the Lord God, “they would deliver neither son nor daughter; they would deliver only themselves by their righteousness.”

Salvation is personal, and will not cover another who has the capability of understanding and choosing.

The verse quoted in the question likely relates to the fact that if you as the head of the household accept God, chances are high that rest of your household will follow your step, provided you act faithfully. Abraham accepted God's call, and by his faithfulness, the rest of his household believed and were followers of God.

It is clear that the verse cited, Acts 16:31, is a promise. The question is, to whom does the promise apply? Taken in context, it is a promise to a particular individual in a specific instance. This verse is lifted from the narrative of Paul and Silas in jail in Phillipi, to which they had been delivered by the owners of a slave girl from whom they had driven an unclean spirit. In the narrative, there was an earthquake during which the doors of the jail came open, and the jailer was about to kill himself, believing that the prisoners had escaped. But Paul told the jailer that the prisoners were all accounted for, and in response to the jailer's question, "what must I do to be saved", Paul responds as quoted. The jailer took Paul to his house, and the jailer and his whole household were baptized.

So yes, the statement of Paul is a promise, and contains, in my view, a bit of prophecy about immediately following events. The jailer believed, and the whole household was baptized. However, it is less clear that this a generalized promise to all mankind, that upon believing, the whole household of the believer will be saved. Were this true, it would put Paul against the prophet Micah, and Jesus, who taught about intra-familial discord between sons and fathers, daughters and mothers, daughters-in-law and mothers-in-law, etc. (cf. Micah 7:6, Matthew 10:21, Matthew 12:34ff, and Luke 12:52-53).