Continuity mistake: Norby rips a bit of Veeko's pants from the right leg; when we see a close-up of Pantoliano, the filament popping in frame from above comes from the left side (if it's a shoelace and not part of the torn pants, Veeko's boots were not unlaced). As he falls, we see for a moment Veeko's feet and there's no torn fabric dangling. (01:13:50)
Continuity mistake: Despite making the jump with Eddie from one building to the next, the two goons come out of the original building they jumped away from. (00:27:00 - 00:28:20)
Continuity mistake: When the two kidnappers realise the baby is drooling on them from the skylight, Mantegna in the shot from above is holding the pen in his right hand. It was in his left a moment earlier. (00:25:10)
Continuity mistake: When the parents finally do find the baby, in the close-up of Bink you can see the left arm of the old soldier who found him first; he was leaning on the armrest at an angle before, but he's resting perfectly horizontal now. (01:26:45)
Other mistake: As Bink's dad looks at the ransom note, the note says "Mr. Cotwell, we have your baby. Put 5 Million $ in a trash bag in the alley behind the Ovaltine factory. No police." Then there are dramatic zoom jumpcuts on parts of the letter, ending with "By midnight." But that last part was not in the note at all. (00:15:00)
Plot hole: There is a scene setting up the nonsensical resolution of the plot, when the policeman recaps the various baby sightings. First; he already got 'fake' reports he chased, there was a scene about that, let's ignore the fact that certainly he would have gotten more like that and he conveniently for the plot lists just the few relevant ones. Second; the baby was kidnapped at 10 and the baby sighting on the bus happens "in the morning." Then the zoo sighting happens "at 3 o'clock." The construction site closes at 5 o'clock. Even forgetting the superior stamina of the 9 months baby that has been walking a whole day without a break, that's a lot of stretch for the actions as shown in the movie, in particular the time between the bus and the zoo; the bad guys would have been loitering in search of the baby for hours between the bus incident and the moment when he leaves the department store to get to the zoo.
Other mistake: Veeko has a hard landing on top of the elevator at the construction site. It's shown when the camera pans up to him, that the elevator doors are vertical scissor gates that open sticking out of the cabin, we see the actual door going up and being lowered to close the gates. Few moments later he is crawling with his body between the cabin and the floor and yet new workers get into the elevator without pushing the gate door up into his stomach. (01:15:55)
Other mistake: The girder is being already lifted when Veeko finds him; he has still to talk to his buddies outside, and they need to get into the construction site, look at him. Before they go to the elevator and take it, more than a minute has elapsed but the girder has gone up just a handful of storeys. (01:09:20 - 01:10:15)
Other mistake: Veeko flips away from the descending elevator, is shown falling almost vertically on his back, and yet in the close-up behind him there's no elevator pit/outside of the building, but a solid inner part of the floor with people doing some construction work. (01:16:20)
Revealing mistake: At the beginning of the three scoundrels' efforts to retrieve the baby from the cage, Brian Haley manages to lean into the cage so deeply he's in with the whole waist, with enough room for Joe Pantoliano to reach around and helping, even. The setup is so absurd that it is scrapped in the rest of the scene, and Veeko can lean in for a much smaller portion of his body (and still can reach the baby just fine). (00:47:35)
Continuity mistake: Baby Bink spots a donut left on the girder by the construction worker. The donut turns around (going by the bitten part) and moves (going by the way it sits on the napkin and how much of it is on one side or the other) between shots, repeatedly. (01:08:30 - 01:09:15)
Revealing mistake: Brian Haley's stuntman launches himself out of the building by his own will in the sequence when the cart reaches the edge (he lets go off the cart handle before the impact and jumps right into 'the void'). (01:17:10)
Continuity mistake: Eddie reaches the "little ball of grief" and promises he's not gonna crawl away. The baby has conspicuously shown drool on his chin as he coos in response, but it's gone a moment later. (01:17:45)
Continuity mistake: In the ending, Bink reaches for a book (hint of the sequel never made, "Baby's trip to China"). The book is sticking out more in the close-up, easier for the baby to grab.
Continuity mistake: When the construction site guard wakes up, in the aerial shot he's in the sun, while in the close-up is in shadow. (01:22:25)
Revealing mistake: In pursuit of the bus, Eddie is turning the van to the left from the main street into an alley. Two different takes have been edited together, as the position of the cars behind and in front of the van changes (not dramatically though, the coordination in the scene has actually been better than much easier scenes). Also you can see on the asphalt obvious skidmarks from previous takes. (00:30:45)
Continuity mistake: The three guys look at the bus leave; Norby is biting his fingers and Veeko has initially his hand on Eddie's shoulder but (same shot) lowers it. Cut to the bus steering, and now Norby has both hands on his thighs and Veeko's hand is back on Eddie's shoulder. While technically they could have time to move their arms like that, they are meant to be "frozen" in shock in the same position and there's no logic in the change. (00:29:45)
Continuity mistake: After the credits, Cynthia Nixon is reading the story to the baby. When there's a transition between an establishing shot of the room and a close-up as she says "Baby Boo's nanny Henrietta says", her long bang in front of the left shoulder disappears. And a moment later, her right hand goes from the side of the book to the middle. (00:03:20)
Continuity mistake: Look behind Norby when he pries open the bus doors; a woman with a teal pantsuit is not exactly inconspicuous as she strolls along the sidewalk. A few cuts after (when he says "A little guy, about two feet tall") the same extra is recycled, walking in the exact same direction. (00:31:50)
Answer: It might have taken a while for them to find the keys to the crane as well as figuring out how to operate it. Once they did, they could get him down.