It is not permissible for a person who does not trust their ability to maintain restraint to kiss their spouse while fasting. This is because such an act may lead to the stirring of desire, to conduct that invalidates the fast, or to seminal discharge. If such a discharge occurs as a result, the fast is invalidated. In this case, the individual continues the fast for the remainder of the day but must make up (qaḍāʾ) that day after Ramaḍān.
As for a person who does trust their ability to maintain restraint, there is no objection to their kissing their spouse. Indeed, ʿĀʾishah (may Allah be pleased with her) reported that the Prophet ﷺ would kiss his wives and spend time with them while he was fasting.
(Ibn Mājah, Ṣiyām, 19; al-Muwaṭṭaʾ, Ṣiyām, 13)