As far as I know it is not possible to export it to excel or pdf.
You might use your browser to export the page for printing, but there might be layout issues depending on the version you are using, as described in this bug.
It is also possible to print the issues as cards, as described in - https://confluence.atlassian.com/display/AGILE/Printing+Issues+in+Card+Form
I hope this information helps.
You can print the JIRA Software gadgets using the PDF View Plugin, including also the Kanban board gadget:
Also check out this sample PDF: http://www.midori-global.com/downloads/jpdf/jira-dashboard-kanban-report.pdf
(Disclaimer: I am a developer of this paid app.)
We use a slightly fiddly but serviceable (and free!) workaround.
What we do is use the appropriate filters to get the board as we want in the browser (we use Chrome) then print to PDF using PDFCreator (there are plenty of other similar programs) as a landscape A2 document.
The 4 columns we use don't quite perfectly stay in line with their issues, but they are certainly close enough that you can tell which issues are in which column without needing to think about it.
We then print it as A3 "to fit" so that it shrinks to fill out a landscape A3. A4 would be too small to be usable for our 4 columns, but if you only use 3 it may be acceptale.
This worked pretty good for us. We have 6 columns in our Kanban, I printed to PDF with size A3 then the default on my printer settings are letter and fit. It printed pretty well except for two aspects: 1) the final column is not included but it is not critical for incomplete work; 2) the column headers overlap the top card in each column (see sample below).
But this is definitely better than printing directly to the printer. I am using Chrome; these results may vary in other browsers.
You can print issues to PDF via Agile Issue Printer plugin (https://marketplace.atlassian.com/plugins/com.flyingagile.jira.plugin.aip)
It works for JIRA and JIRA Agile so you can easly print issues in selected psrint or event Sprint state column
Excel would not be a very good option for this, I'm affraid. One thing you could try is implementing the Kanban board layouting logic using the PDF View Plugin.
Then you have a representation optimized for printing. I'm not saying it is trivial, but definitely doable.
Start learning here:
(This actually sounds very interesting, so please free to reach out to us at info at midori.hu any time with your questions or results!)