You seem to have several related questions here, so i'll try and address them all.
Is it true that the Messiah will be a descendant from the house of David?
Yes, it is. Maimonides, a 12th-century rabbi who was one of the foremost codifiers of Jewish law, says so in his Mishneh Torah, Laws of Kings and Wars 11:4:
ואם יעמוד מלך מבית דוד ... הרי זה בחזקת שהוא משיח
If a king will arise from the House of David ... [who fulfills several conditions] ... we may, with assurance, consider him Moshiach. [translation from kesser.org]
This is quite basic and is accepted by everyone (TTBOMK).
Do we even know the descendants of the house of David?
Some people have claimed to be his descendants. We have no way of knowing for sure though, besides for genetic testing (and i don't think we have David's genes to compare against).
Should we then not keep a close watch on every human birth of every baby who are a descendant from the house of David? Should this not be a priority?
No. First off, we don't really have conclusive proof of who is and who isn't of the House of David. Second, watching the babies won't really help. Mashiach (the Messiah) will reveal himself when it is time through his actions. Continuing the quote from Maimonides at the beginning of the answer:
ואם יעמוד מלך מבית דוד הוגה בתורה ועוסק במצות כדויד אביו, כפי תורה שבכתב ושבעל פה ויכוף כל ישראל לילך בה ולחזק בדקה ויילחם מלחמות ה' הרי זה בחזקת שהוא משיח אם עשה והצליח וניצח כל האומות שסביביו ובנה מקדש במקומו וקבץ נדחי ישראל הרי זה משיח בודאי
ואם לא הצליח עד כה, או נהרג--בידוע שאינו זה שהבטיחה עליו תורה, והרי הוא ככל מלכי בית דויד השלמים הכשרים שמתו.
If a king will arise from the House of David who delves deeply into the study of the Torah and, like David his ancestor, observes its mitzvos as prescribed by the Written Law and the Oral Law; if he will compel all of Israel to walk in [the way of the Torah] and repair the breaches [in its observance]; and if he will fight the wars of G-d; - we may, with assurance, consider him Moshiach.
If he succeeds in the above, builds the [Beis Ha]Mikdash on its site, and gathers in the dispersed remnant of Israel, he is definitely the Moshiach.
- If he did not succeed to this degree or he was killed, he surely is not [the redeemer] promised by the Torah. [Rather,] he should be considered as all the other proper and legitimate kings of the Davidic dynasty who died.
So watching babies wouldn't help, as there's no way to know until he fulfills his duties and is crowned king.
See also, (linked by @Danno in the comments), JewFAQ on Mashiach.