RS232 is +-12V, you could cause damage by connecting this directly to an arduino. To convert these voltages you need a special chip, such as the MAX 232 to convert from RS232 to 5V TTL (MAX232 - Wikipedia).
By the sounds of things you don't need an arduino, just either a computer with a serial (RS232) port, or, a USB-RS232 converter cable (equally a FDTI or equivalent cable with the MAX 232 style converter). Then I'm sure you'll be able to find some open source software which will run on whatever OS you use, with which you can log the fluorometer data.
If on the other hand you would prefer to have a non-pc version, then yes the arduino and a datalogging shield would work, you could log to SD card in any format you so wish. Just remember to convert to TTL level!