For a fish to be kosher, it needs to have both fins and scales, and the scales must be easily removable without tearing the skin.  So although many bottom dwellers such as shellfish, lobster, crab, catfish, etc., aren’t kosher, it’s because they don’t have the signs of a kosher fish.
Carp, on the other hand, is often classified as a bottom feeder, and is one of the more popular kosher fish used in making the classic “gefilte fish” for Shabbat.
Swimming to the Top
Torah laws can generally be classified into different categories: those that are logical and those that transcend human logic. Kashrut, for which no reasons are given in the Torah, falls into this second category.
At the same time, our sages tell us that we are meant to delve into these laws and endeavor to comprehend them as much as possible (all the while keeping in mind that the “ultimate reason” behind these laws may be beyond our limited human intellect). 
Thus, many rabbis and philosophers have speculated on their purpose.
For example, Rabbi Moshe ben Nachman (Nachmanides) writes in his commentary to Leviticus:
“The reason why fins and scales [are signs of permissibility as food] is that those fish which have them always dwell in the upper, clear waters, and they are sustained through the air that enters there. Therefore their bodies contain a certain amount of heat which counteracts the abundance of moistness [of the waters], just as wool, hair and nails function in man and beast. Those fish which have no fins and scales always dwell in the lower, turbid waters, and due to the great abundance of moistness and gatherings of water there, they cannot repel anything. Hence they are creatures of cold fluid, which cleaves to them and is therefore more easily able to cause death, and it [the cold fluid] does in fact cause death in some waters, such as stagnant lakes.” 
This seems to be one of the main sources for the notion that all kosher fish are by definition not bottom feeders. However, as noted, this is only meant as an “after-the-fact” observation, not an absolute rule.
Additionally, a closer look at Nachmanides reveals that it’s not about where the fish feed, but whether they swim to the upper levels of the water.
Interestingly, carp, which is often classified as a “bottom feeder,” actually feeds in nearly all sections of the water, and feeds on the bottom only 40–50% of the time. They are also known to feed directly on the surface.
Perpetuating the Misconception
Confusing the speculations of philosophers with “the” reason for kashrut has led to many misconceptions, the most common of which is that kosher food is specifically healthier.
True, due to some of the kosher guidelines as well as the strict controls put in place by some kashrut organizations, kosher products may be healthier, but this is not always the case. To illustrate, hemlock, one of the deadliest poisons, comes from a plant and is completely natural, and therefore is technically “kosher.”
Of course, one of the commandments in the Torah is to be careful with our health,  so we shouldn’t eat anything that is harmful to us. Yet what is harmful for one person may be fine, or even healthy, for another. Therefore the question of health is left to doctors (who themselves are not always in agreement), and the question of kashrut is left to rabbis.
So although the common misconception is that bottom feeders are unhealthy and therefore unkosher, their health status doesn’t impact their kosher status. (And some bottom feeders are in fact healthy.)
Although the underlying reason for keeping kosher isn’t necessarily physical health, the sages and mystics explain that it does have a profound effect on one’s spiritual health and wellbeing. Furthermore, eating non-kosher foods can dull one’s spiritual sensitivities. And like all things in the spiritual realms, this can in turn impact one’s physical wellbeing as well. 
Kosher, Healthy Mind and Body
Interestingly, the Talmud states: “All [fish] that have scales also have fins [and are thus kosher]; but there are [fish] that have fins but do not have scales [and are thus unkosher].” 
The Lubavitcher Rebbe explains that scales, which protect the fish like armor, represent the quality of yirat shamayim, “awe of heaven” and integrity, which protects us from negativity and the many pitfalls that life presents. Fins, which propel a fish forward, represent innovation and ambition.
Ambition and innovation without the protection of integrity and awe of heaven can have very negative consequences, and isn’t “kosher.”
At the same time, we can’t just remain in a protected state—in matters of holiness, we must always strive to move forward and climb ever higher. 
 Leviticus 11:9–12 and Ramban ad loc; Deuteronomy 14:9–10.
 See Mishneh Torah, Hilchot Temurah 4:13.
 Ramban, Leviticus 11:9. Rabbi Moses Sofer, known as the Chatam Sofer (on Talmud, Chulin 66a) explains that Nachmanides’ intention is that fish with fins and scales can propel themselves to the top of the water where there is more air and warmth, enabling them to produce scales. The scales provide a method for the fish to get rid of some of the excess harmful moisture that would otherwise be trapped in the fish.
 Deuteronomy 4:15; Talmud, Shabbat 32a.
 See Talmud, Yoma 39a; Rabbeinu Bechayei, Exodus 23:19 and Leviticus 11:43; Tanya, chs. 7–8.
 Talmud, Niddah 51b.
 Reshimot, no. 39.
Source: Yehuda Shurpin – CHABAD