Do Defendants in civil matters (in the United States, and in New York
specifically) have a right to court transcripts if they cannot afford
it?
No. They do not (with, as always, certain exceptions).
There may be an exception in the case of termination of parental rights proceeding (where there is a right to counsel), and in quasi-criminal contempt of court proceedings (where there is sometimes a right to counsel).
There is a U.S. constitutional right to free transcripts for an indigent criminal defendant in connection with a first appeal of right if state or federal law provides for a direct appeal of right (which it almost always does, even though there is no constitutional right to a direct appeal of a criminal conviction). In some cases, a federal court in a habeas corpus case collaterally attacking a state court criminal conviction can order a free transcript (per the same link), even though this is technically a civil proceeding.
What are Joe's options?
This said, while a transcript is normally required for an appeal, unless the legal error is apparent on the face of the judge's order, a transcript is normally not required, even if it is preferred, for a post-trial brief. Joe can still reference what he recalls happening at trial and reference the relevant exhibits.
Also, in a dissolution of marriage proceeding, a court has great discretion and authority to order the more affluent spouse to pay all or some of the legal fees and/or costs for the less spouse in the case, even as part of temporary orders in a case. This is not a sanction and is not based upon the merits of the arguments of the parties. It is simply a reflection of the fact that each spouse has some inchoate interest in the other spouse's property in marriage that becomes manifest at the time of a divorce. Joe could ask the judge to have his spouse pay for some of his legal fees and/or costs, such as the cost of a trial transcript, in the case.
This is not a constitutional right. It is discretionary authority afforded to the judge as part of the state's substantive law related to the dissolution of marriage. Specifically, in New York State, this is a consequence of New York Domestic Relations Law § 237(a). This states, in the pertinent part (emphasis added):
In any action or proceeding brought . . . (3) for a divorce . . .the
court may direct either spouse . . . to pay counsel fees and fees and
expenses of experts directly to the attorney of the other spouse to
enable the other party to carry on or defend the action or proceeding
as, in the court's discretion, justice requires, having regard to the
circumstances of the case and of the respective parties. There shall
be a rebuttable presumption that counsel fees shall be awarded to the
less monied spouse. In exercising the court's discretion, the court
shall seek to assure that each party shall be adequately represented
and that where fees and expenses are to be awarded, they shall be
awarded on a timely basis, pendente lite, so as to enable adequate
representation from the commencement of the proceeding. Applications
for the award of fees and expenses may be made at any time or times
prior to final judgment. Both parties to the action or proceeding and
their respective attorneys, shall file an affidavit with the court
detailing the financial agreement between the party and the attorney.
Such affidavit shall include the amount of any retainer, the amounts
paid and still owing thereunder, the hourly amount charged by the
attorney, the amounts paid, or to be paid, any experts, and any
additional costs, disbursements or expenses. An unrepresented
litigant shall not be required to file such an affidavit detailing fee
arrangements when making an application for an award of counsel fees
and expenses; provided he or she has submitted an affidavit that he or
she is unable to afford counsel with supporting proof, including a
statement of net worth, and, if available, W-2 statements and income
tax returns for himself or herself. Any applications for fees and
expenses may be maintained by the attorney for either spouse in his or
her own name in the same proceeding. Payment of any retainer fees to
the attorney for the petitioning party shall not preclude any awards
of fees and expenses to an applicant which would otherwise be allowed
under this section.
Note also that this provision does not depend upon one's status as a Plaintiff or Defendant (and in divorce cases, they are more commonly called a Petitioner and a Respondent, or a Petitioner and a Co-Petitioner, as the case may be).
