Can the cursor reach the bottom?
APL (Dyalog Unicode), 60 bytesSBCS



Full program. Prompts for character matrix from stdin. Prints 1 or 0 to stdout. Requires zero-indexing (⎕IO←0) which is default on many systems.

P←{' '0∊⍨s←(×⍨≢⍺)⊃,⍵:1∊⍵⋄s}
1∊⊢⌿{P⌺3 3⊢(⍴⍵)⍴1@0P⌺3,⍵}⍣≡3/3⌿⎕


Try it online! (space characters replaced by ░ for clarity)

The method here is basically to seed fill from the top left corner, but alternating between 1D and 2D seed fill to account for line wrapping and vertical/diagonal motion. Diagonal motion is simple horizontal motion followed by vertical motion, since the cursor sits in between adjacent characters.

The first line defines a helper function P(rocess) which will be called on each neighbourhood.

P←{…} "dfn"; ⍵ is the neighborhood, ⍺ is the number of padded elements for each dimension:

 ,⍵ ravel (flatten) the neighbourhood into a list

 (…)⊃ pick the following element from that:

  ⍺ the paddings per dimension (so a 1 or 2 element list, depending on ⍵ being 1D or 2D)

  ≢ the tally of that (1 or 2)

  ×⍨ the square of that (lit. multiplication selfie; 1 or 4)

 s← assign to s(elf; i.e. the centre of the neighbourhood)

' '0∊⍨ is that a member of [" ",0]?

 : if so:

  1∊⍵ return whether the neighbourhood has a one

 ⋄ else:

  s return s(elf unmodified, i.e. leave as-is for now)

The second line is the main part of the program:

⎕ prompt for evaluated input via the console (stdin)

3⌿ triplicate each row (ensures we have at least 3 — makes no difference to connectivity)

3/ triplicate each column (ensures we have at least 3 — makes no difference to connectivity)

{…}⍣≡ apply this function repeatedly until two successive results are identical (i.e. until stable):

 ,⍵ ravel (flatten) the matrix into a list

 P⌺3 apply P to each neighbourhood of 3 elements

 1@0 replace the current value at position 0 with a 1 (this seeds the top left corner)

 (…)⍴ reshape that into the following shape:

  ⍴⍵ the original shape of the matrix

⊢⌿ get the bottom row (lit. vertical right-reduction)

1∊ is there any 1 there?