The Shearim Metzuyanim Behalacha on the Kitzur Shulchan Aruch (חלק א - כג ס"ק א, מהדורה תשס"א) writes:
As is known, we are obligated to stand for the Sefer torah, and this is learned from the obligation to stand for a torah scholar, for if we stand for those who learn the the Torah, all the more we should stand for the Torah itself. Rashi in Avoda Zara (17a) states that it was the custom when taking leave from the Synagogue, the people would kiss the most honored among them in a sign of respect, following this it would be fitting to kiss the Sefer torah as well, in line with Kal Vechomer the Gemara makes to obligate standing for the Torah from the obligation to stand for the Talmid Chacham. This would seem to be the source for the custom to kiss the Torah scroll as it passes.
This is the interpretation of the Rema (או"ח קמט) that we bring the children towards the Torah to kiss it on post to encourage and educate them in the observance of mitzvos, that this does not mean only children, rather even adults should kiss it but that there is a value in educating the children to this as well.
The Siddur Yaavetz and other siddurim quote the custom of the Ariza"l was kiss the sefer torah, it is also said that one should kiss with his lips specifically, and not the hands as is the custom of the general public. However the Sha'ar Ephraim (שער י' ס"ד) says that if it is difficult to kiss with the lips one can kiss with his hand.
The Pninei Halacha (מדיני קריאת התורה פרק כב' ס' ג) writes:
נהגו שכל מי שספר התורה עובר לידו מנשקו ומלווהו מעט. הרוב נוהגים לנשק
את התורה בפיהם ממש, ויש שנוגעים בו בידם ומנשקים את היד (ע' פס"ת קמט,
א-ב). וראוי שחולה או מצונן לא ינשק את התורה בפיו, כדי שלא להדביק את
שאר המתפללים במחלתו.
The custom is that anytime a Sefer Torah passes one should kiss it and accompany it a bit.
The majority have the custom to kiss the Torah directly with their lips, but some touch with their hand and kiss the hand.
It is preferable of someone is ill should not kiss the Torah with his mouth so as not to infect others with his illness.