Hello,

I've been riding my Papillionaire Cafe Racer about seventeen miles a day along the Mississippi River path, and down onto St. Charles Avenue in New Orleans. The bike's got the basic thin aluminum rims and tires, and I've been considering getting fatter tires to lessen the chance of flats while on the street. Riding in the bike lane is fine, but when you have to pull close to the curb, you run into all sorts of gravel, broken glass, etc. I'm wondering if wider tires would put undue pressure on the walls of the rims and ruin them. Would I have to get wider rims, or would my existing rims be okay for balloon tires?

Any advice would be appreciated.

First question you should ask if true balloon tires will fit your frame. I have several bikes with balloon tires like Schwalbe Fat Franks (both 26 and 29) and Big Apples (in 29x2.35) and they are awesome for general riding and gravel/broken chipseal roads. I also do not hesitate to ride them on dry packed singletrack.
They swallow all road imperfections

Tire liners. Many people hate 'em, I love them. I commute on three bikes and the goat head thorns out here were causing a flat every few days. Now, maybe one a year.

the bike comes with 28mm tires.
First thing you have to do is check your clearances - height to the brake and side to the fork.
Fewer bikes are limited by rear brake, and almost all are limited by chainstays, so figure out first what will fit.
You very likely have clearance for 32mm, maybe 35mm, and you're if you're really lucky, 38mm.

You don't have to have bulletproof tires to resist flats.
I run the lightest weight tires I can get in big sizes, run them at low pressures (+latex tubes), and for twice as many miles, I flat about 20% of my buddy who runs city tires on 3 bikes. You're probably not going to want to spring $76/tire for Compass ELs.
Look at some of the new gravel grinder tires. They're good both on pavement and off, many are light, designed for low pressure and pretty tough.
Gravel Kings are great choice.

I run my 38mm Barlows on narrow Open Pro rims, so you don't have to worry about the rims - check clearance and decide how much room you have for bigger tires.

Always good to carry a spare tube and a pump. Patch the other tube later at home.
If you want a really simple flat kit, and have removable-valve-core tubes, a 2-oz bottle of Stans with with the valve core tool and a pump will get you home, and you don't even have to remove the wheel. I give some credit to my latex tubes, but every time I've used Stans for a roadside flat, the Stan's repair as lasted as long as the tire tread.

"Balloon tires"? What is this, 1933? Do you wear a raked fedora and a double-breasted suit, cigarette dangling from your mouth, and call all the women "dames"? You want them whitewall, as well?

FWIW, increasing the width of your tire will only increase the chance of actually hitting a piece of glass. Money would be better spent on city-specific tires.

Whitewalls is a good idea, besides esthetics it could be a safety feature too.

It's not 1933, but it isn't 1983 either. The idea that thinner is faster wasn't very scientific to begin with, and views have changed. In general the 30's have produced much more good idea's for commuter bike's than the 80's and most fat tyres don't go flat very easily.

Fatter tires are also generally run at lower pressure, allowing more flex and give before a puncture occurs.

That said, I don't disagree that the OP may be better served by a puncture resistant tire.

If you want to reduce your chance of flats, get tires that are specifically meant to reduce flats, like the Schwalbe Marathon Plus:
Marathon Plus HS 440 | Schwalbe North America

That's the best tire you can get (others may be similar but no one is better) for an existing bike for flat resistance.