They wrote their names in English (or "Federation Standard"*), because if they hadn't, Sisko wouldn't have been able to read it.
I have personally been in the situation of being asked, while in a non-English speaking country, to write a note or card to a local. I got a local friend to help me write in the local script. Likewise I've had foreign friends ask me to check their cursive English handwriting on a birthday card to make sure it is readable. It's just a nice gesture. Even if the DS9 crew aren't fluent in "Federation Standard", the fact that they tried makes it an even more meaningful gift to Sisko. It certainly makes sense for Kira her to try and make her name readable as a token of respect to the Emissary.
Another possibility is that Kira/Quark (and Odo?) had someone write their names for them. This might explain the ears on Quark's name as someone's attempt to be cute.
Worf should definitely be fluent in Earth handwriting.
Regarding "Yet we know that on DS9 the Universal Translator is in constant use", well that may be true but we that's also a contrived plot device. The popular idea among fans that everyone is really speaking their own language all the time and we just hear English isn't backed up by evidence, as pointed out in the accepted answer to this question (despite some episodes seeming to contradict it). We only need assume it's working on aliens that obviously couldn't be speaking English. No need to extend the plot device any further.
*Federation Standard is a non-canon stand-in for English, but we can choose to incorporate this information from Memory-Beta (referencing various EU novels) as evidence that there is a common Federation language with a script people would be familiar with.
When the Republic of Bajor joined the Federation in the year 2376, many signs and directories located throughout the planet were altered to include both Bajoran and Standard directions.