Start with the height you have
Go and measure from the basement floor to the underside of the joists. Not to the drywall, to the joists. Then measure again under a duct, because that is the number that decides how the room feels.
Under about 6 feet 6 inches, the basement will always feel like a basement no matter what you spend on it. Around 7 feet it starts to read as a room. At 8 feet it reads as part of the house.
Most Toronto houses built before the sixties give you somewhere between 6 and 7 feet. That gap is the entire reason anyone underpins.
What underpinning actually is
The footing under your foundation wall sits at a certain depth. Underpinning goes below it, in short sections, and extends the wall and the footing downward. You dig out a metre or so of the basement floor, and the walls follow the floor down.
It is done in sequence, never all at once. A crew excavates and pours one section, lets it cure, then moves along the wall. The order and the widths matter enormously, because at any moment part of your house is being held up by the sections either side of the one that is open.
That is why this is not a job for a general handyman with a bobcat. It needs an engineer's drawings, a permit, and a crew that does this specific work. When underpinning goes wrong it does not go slightly wrong.
Benching is the cheaper answer nobody explains
Benching leaves the footing where it is and builds an angled concrete shoulder inward from the base of the wall, then drops the floor in the middle.
You get the ceiling height, in the middle of the room. What you lose is floor area around the perimeter, and the bench itself is a permanent concrete ramp along your walls that you have to design around.
Benching is meaningfully cheaper and faster than underpinning, and there is no engineering risk of the same order. In a wide basement where the perimeter is not precious, it is often the right answer and it rarely gets offered, because it makes for a smaller invoice.
Ask for both prices. If a contractor will not price benching, that tells you something.
What it costs
Underpinning in Toronto is generally quoted by the linear foot of wall. The rate moves with access, soil, how deep you are going and what is in the way, so the honest figure is a range rather than a number: most residential jobs land somewhere in the two to five hundred dollars a linear foot band, and a typical semi-detached perimeter is a lot of linear feet.
For a whole basement, expect the underpinning alone to be a five figure number well into the middle of that range, before you have a finished room. Then add the things that always come with it: a new floor slab, new drainage and a sump, waterproofing while the walls are exposed, and usually the electrical and plumbing rough-in.
Benching commonly comes in materially under that, and the saving is mostly in engineering, sequencing and time.
Neither number includes finishing the basement. That is a separate project and it is the one people budget for and then discover they cannot start.
Do it while the walls are open, or do not do it at all
If you are underpinning, everything that belongs against or under that wall gets done now.
Interior weeping tile and a proper sump. Exterior waterproofing if you are excavating outside anyway. A new slab poured over vapour barrier and rigid insulation rather than straight onto soil. Plumbing rough-in for a bathroom you might want in three years, roughed in now while the floor is open.
Coming back later to do any one of these means opening what you just closed. The marginal cost during the dig is small. The cost afterwards is the whole job again.
The honest test for whether you should
Underpinning pays for itself in two situations and neither of them is "we wanted a nicer basement".
The first is when the extra height creates real, usable space in a house that has run out of it. A family that needs a fourth bedroom and cannot afford to move is a good candidate. Height turns unusable square footage into house.
The second is a legal secondary suite. A basement apartment has to meet minimum ceiling height, egress, fire separation and a set of other requirements before anyone will call it legal, and height is the one you cannot solve any other way. If a rental income is the plan, the underpinning is not a luxury, it is the precondition.
Outside those two, look hard at benching, and look harder at whether the money would do more upstairs.
How long
An engineered drawing set and a permit runs a few weeks to a couple of months depending on the municipality. The underpinning itself, for a typical Toronto house, is usually several weeks of sequenced work, and it does not compress. The sequence is the safety.
Then the basement is a shell, and finishing it runs like any other build of that size.
Plan on the whole thing being a season of your life. Anyone quoting a fortnight is quoting one wall.