This simple verse shows yet another minor detail which appears to be correct in the Book of Mormon. It is a small thing to say that they crossed the river, but were this a concoction of Joseph Smith, why should it be mentioned?
Given the logical path of the family along the Eastern borders of the Red Sea a Wadi would be created when the water ran from the higher level to the lower level, which of course is the Red Sea. From the topography of the area, it is far more likely that a river would run in an approximately perpendicular course to the Red Sea, and thus create a temporary barrier to their crossing. They crossed the River Laman because there was no other way.
The crossing of the river also tells us just a little more about the nature of the river. We know that "rivers" in that area are seasonal, and that in the season of their arrival water was coursing down the wadi. Since there is no mention of building a bridge, it may be assumed that fording the river was no task of great consequence, and therefore may be assumed to be shallow, and at the time of their crossing, not overly swift. This description also fits a river which would be created from large and sudden sources of water, which might begin quickly, and then slow and begin to fade. It is also possible that the river was coursing much deeper and faster when they arrived.
Of course this brings up Lehi's poetic comparison of the river and his son: "1 Nephi 2:9 And when my father saw that the waters of the river emptied into the fountain of the Red Sea, he spake unto Laman, saying: O that thou mightest be like unto this river, continually running into the fountain of all righteousness!" If the river was indeed the periodic rivers which flashed down the wadi's, why does Lehi use the term "continually running into the fountain...."
First, the term "fountain" is used in Lehi's vocabulary to mean a standing body of water, which in the case of the concrete example was the Red Sea, and in the case of the gospel image, the gospel itself. In both cases the "fountain" is a constant, and is always there. How is it that the river is "continually running?"
Whether or not water is present, the water course is set. The wadi which delivers the water to the Red Sea is the low point into which water naturally flows. It is the permanent conduit of the river. It is not improbable in the world to find rivers which periodically do not flow, but which are yet known and named during the dry periods. There are parts of the Rio Grande in the United States which may be dry at times, but the course is know as the Rio Grande, whether or not the water is present. It is the river course that channels the water which is continually running. As long as water is present, the water will run to the "fountain."