This question is hard to answer in detail, but understanding the concept should give you a pretty good idea how to approach the matter.
The Torah writes, “When you build a new house, you shall make a fence for your roof, and you shall not place blood in your house, should the one who falls fall from it.”[1] Although the pasuk mentions just a roof, Chazal[2] extended this law to a variety of dangerous places (such as a pit) in one’s property. You refer to a guardrail for a staircase, which can be a dangerous place, especially for small children.
If the halacha applies to all places where a person can fall, then why is the roof singled out? There are a few basic approaches one can take to answer this question. The Sefer HaChinuch[3] says that the Torah just gave a common example of a place that requires a fence. However, there is another, not necessarily contradictory, approach found in several Acharonim, which seems logically appealing according to classical halachic analysis.
Let us employ an analytical tool that some of us like to call “tzvei dinim” (Yiddish), two elements to the halacha. The requirement of a fence for a roof is quite objective and formalistic. The requirement elsewhere is more subjective, and it depends on the specifics of the situation. This distinction makes the case of a roof often stricter but, at times, more lenient than other cases. For example, a dwelling that doesn’t meet a house’s size requirements is exempt from having a fence for its roof even if it is used in the same manner as other roofs.[4] Additionally, the minimum height of the fence is ten tefachim (roughly two and a half feet), hardly enough to totally prevent someone from falling.[5] Rather, this height is the classic one for a halachic wall in a variety of contexts, from the laws of sukka to the laws of eiruv and elsewhere. Thus, the maximum space in between the vertical bars of a roof’s fence is probably three tefachim (approximately nine inches), the normal size for a gap to be considered halachically significant.
That is in regard to the more formalistic and defined application of these halachot. Nevertheless, by extending the concept to a wide range of dangers (including raising a “bad” dog[6]), Chazal were telling us that, beyond the formalistic element of the mitzva, the spirit of the law is binding as well. Thus, where there is palpable danger, further steps may need to be taken. This requirement is derived not from the positive commandment to “build a fence” but from the negative commandment not to “place blood in your house” and the more general commandment to “be careful and safeguard your life.”[7] One difference that stems from the distinction between the more defined and less defined applications of the halacha is that even those who require a beracha when building a fence[8] do so only for a fence on a roof.[9] Another distinction is that, in other cases, one first must determine whether a potentially dangerous area is actually used.[10] This is in unlike to the law in the case of a straight roof, which needs a fence as long as it can, in theory, be used.[11]
Thus, in your case, one has to consider what the actual dangers are. If there is reasonable danger for children, then you have to ask a technical expert what the maximum width between bars should be. Although halacha does not expect one to spend all of his money removing the most remote danger, it is, in general, better to err on the side of caution.