How do I connect a 5 1/4" floppy drive to a modern PC?
A company called Device Side Data produces a device called the FC5025 which can adapt a 5 1/4" drive to a USB port. You will still need to power the drive so this approach will require either an external power source or a tower or desktop computer. There are also other limitations to this device, such as the inability to write to the disks or to read most forms of copy protection (back when these drives were common there was a technique to protect their contents from copying by writing intentionally bad data).
Using just original hardware the best approach would be to build or buy a computer that had a 5 1/4" inch drive and could connect to a network or had a 3 1/2" drive as well. 3 1/2" drives are still common enough to find as USB external drives, while networking a old PC would be complex.
There is a device available from the Software Preservation Society called a kryoflux which is a USB disk controller and can be used to create image files from many disk formats such as MS-DOS, Apple, Commodore, etc. Unlike the FC5025, the Kryoflux has write support.
It comes with a command-line program and a Java graphical front end for those who prefer to point and click. Linux, Mac, Windows and AmigaOS are supported.
I have used this device to create image (.img) files from old 5.25" MS-DOS disks on my up-to-date i7 computer. It is more involved than a native supported drives of bygone days but it does work. For example, imaging a 360K disk goes something like this:
$ dtc -ftest.img -d1 -k2 -v300 -i4 -e40
which you can then mount
$ sudo mount -o loop test.img /mnt
$ $ ls /mnt
4201.CPI CHKDSK.COM DISKCOPY.COM FDISK.COM JOIN.EXE NLSFUNC.EXE SYS.COM
5202.CPI COMMAND.COM DISPLAY.SYS FIND.EXE KEYB.COM PRINT.COM TREE.COM
ANSI.SYS COMP.COM DRIVER.SYS FORMAT.COM LABEL.COM RECOVER.COM
APPEND.EXE CONFIG.SYS EDLIN.COM GRAFTABL.COM MODE.COM SELECT.COM
ASSIGN.COM COUNTRY.SYS EXE2BIN.EXE GRAPHICS.COM MORE.COM SORT.EXE
ATTRIB.EXE DISKCOMP.COM FASTOPEN.EXE IO.SYS MSDOS.SYS SUBST.EXE
(that example is imaged from a 360K 5.25" MS-DOS boot floppy)
Another device is the Supercard Pro. I have no personal experience of this one but I did find this review and this page which contains a lot of useful information ragrding the Kryoflux, Supercard and other boards and utilities for reading and preserving floppy disks.