The interpretation and implementation of the Word of Wisdom has changed over time, which is why portions of it (like this reference to mild drinks) can be confusing to modern readers. I divide my answer into three portions.
Early Leaders (pre-1900)
As far as I know, there are no statements from early prophets and apostles about their interpretation of the phrase "mild drinks." Historians who do make inferences about their interpretation of the Word of Wisdom do so on the basis of their recorded behavior. Because early leaders of the church drank wine and beer moderately, some infer that they did not consider the moderate consumption of these beverages to be an infraction of the Word of Wisdom. For instance, see this quote from a BYU Master's thesis titled "An Historical Analysis of the Word of Wisdom" on page 26:
Despite the injunction contained in the revelation discouraging the drinking of wine, (except for sacramental purposes) the casual nature of the allusions to this beverage suggest that many Church Authorities did not consider moderate wine drinking in the same category as the use of strong drinks.
See also this discussion from FAIR, a Mormon apologetics group.
In the early 1900s, leaders of the church began to enforce the Word of Wisdom as a commandment. Joseph F. Smith describes this shift as follows, from the October 1913 General Conference (see page 14):
The reason undoubtedly why the Word of Wisdom was given as not by "commandment or restraint" was that at that time, at least, if it had been given as a commandment it would have brought every man, addicted to the use of these noxious things, under condemnation; so the Lord was merciful and gave them a chance to overcome, before He brought them under the law.
At the beginning of this shift, there was some disagreement between leaders of the church over what was or was not allowed by the Word of Wisdom. A paragraph from this article (in Dialogue: A Journal of Mormon Thought, page 78) helps illuminate the fact that some leaders thought beer was not forbidden under the Word of Wisdom:
Though it is clear that some church leaders, like Heber J. Grant and Joseph F. Smith, insisted upon complete abstinence from tea, coffee, liquor and tobacco, all General Authorities were not in agreement on all aspects of the Word of Wisdom. During a discussion in 1900 after he became President of the Church, Lorenzo Snow again emphasized the centrality of not eating meat, a point rarely emphasized by others, and in 1901, John Henry Smith and Brigham Young, Jr., of the Twelve both thought that the Church ought not interdict beer, or at least not Danish beer. Other apostles, like Anthon H. Lund and Matthias F. Cowley also enjoyed Danish beer and currant wine. Charles W. Penrose occasionally served wine.
However, this disagreement was overcome, and the Word of Wisdom began to be enforced in its current form.
The church's current enforcement of the Word of Wisdom clearly forbids drinking any alcoholic beverages, the moderate consumption of wine and beer included. But if you still want to quench your thirst for a mild barley-based beverage, you could grab one of these.
Some helpful links, in addition to the ones in this answer: