I believe there are currently 2 questions being asked. One more general 'Should a websites filters be saved across visits' and 'Should I save the filters on this job website' and perhaps a third how to show it to the user. I will answer the general question first and use it to answer your specific question.
Should a websites filters be saved across visits
Saving filters is not a general Yes / No answer. There are many different websites and the intent of the users might vary.
How do users use the filters?
So, part of the question is: How do users use the filters. Will the filters be applied just once because the user is clearly looking for just one thing or will the user use different filters over time. For instance, look at the screenshot of Amazon below.
As you know Amazon offers many different products, if you need to buy a product from Amazon it will almost always be a different product, perhaps you might buy a book this week and next week a game. In this instance I would not keep the search settings, as the intent of the user will differ each week.
However, Booking.com does save part of my search filters. As you see below here.
However, the website did not save my filters I applied. I use Booking.com fairly often from different devices (Tablet / Mobile phone / Desktop at home & work). I never login, however I still find it annoying that I have to set my filters up each session. As a non registered user Booking.com does not offer that option.
Complexity of the product
So, another part that I deem important is the complexity of the product. Will the user input a lot of filters before finding. In the case of Booking.com I do enter a lot of filters each time. For instance, I always need to find a Hotel that accepts pets because I bring my dogs. Secondly I always filter with something in mind. I nearly always use 3 to 4 filters, so I would find it awesome if they offered me the option to enable my last selected filter(s).
I would find it logical for Booking.com to remember my filters (or offer me a possibility to keep the filters).
Now lets say the thing I want to search for is not that complex and I offer a minimum set of filters (perhaps a total of 3). It will be fairly easy for the user to re-select the filters, it might take a bit longer but it will not 'hurt' their time spend on the last visit. Offering perhaps to enable last filters through a small popup somewhere might me more annoying then helpful.
Now, the reason Booking.com does not save the last filters might be because of privacy. I do believe that a good amount of people view Booking.com from a public computer. I would not find it awesome that the next user that visits booking.com from a public computer would know where I want to go and what filters I used.
Is/can the user logged in
If the user is or can login I would offer an option to save the last used filters, if it is logical according to the problems mentioned above.
Should I save the filters on this job website
Now, I would imagine that you do have a few filters on the job website. A couple I can think of is: Job title, Location + range, Possible benefits, Term duration etc. So this is already fairly complex. The website is already a niche website, you can only find jobs on the website. However, privacy it might be extremely sensitive to keep the data (but you know this better then me). Looking at a large competitor in job listings, Indeed.com. They do show the last used search query/filters.
How show last used filters to a user
As this is more of a side question, I will keep this short. Personally I would find it best if there was some kind of small box that I just have to click to get my last filters (for instance like Indeed.com did).